LIBRARY 

OF  THE 

University  of  California. 


GIFT    OF 


I 


e^^^f^J^^^U, 


A^;a^^ 


APPLETOJfS'  MATHEMATICAL  SERIES 


MJMBERS  SYMBOLIZED 


AN 


ELEMENTARY  ALGEBRA 


BY 

DAVID   M.  SENSEMG,  M.S. 

PROFESSOR  OF  MATHKMATICS,   STATE  NORMAL  SCHOOL,   WEST  CHESTER,   PA. 


NEW  YORK,  BOSTON,  AND  CHICAGO 
D.   APPLETON    AND    COMPANY 

1888 


St 


Copyright,  1888, 
By  D.  APPLETON  AND  COMPANY. 


PREFACE. 


The  aim  of  this  volume  is  to  lay  the  foundation  for  a 
more  extensive  and  philosophical  treatise  soon  to  follow, 
and  to  aid  in  supplying  the  needs  of  the  common,  high, 
normal,  and  other  preparatory  schools  and  academies, 
where  the  time  allotted  to  this  department  of  knowledge 
is  necessarily  limited  to  an  elementary  treatise. 

In  scope  it  includes  all  subjects  essential  to  a  study 
of  higher  arithmetic,  elementary  geometry,  and  the  ele- 
ments of  physics.  All  matter,  however,  is  treated  in  an 
elementary  manner,  so  that  any  ordinarily  intelligent  stu- 
dent, with  a  fair  knowledge  of  the  principles  of  common- 
school  arithmetic,  may  master  it.  All  broad  generalizations 
and  discussion  of  general  problems  have  been  purposely 
excluded. 

In  the  earlier  lessons,  fundamental  ideas  and  principles 
are  developed  inductively,  and  then  formulated  into  as 
simple  and  concise  statements  as  is  consistent  with  truth. 
Further  on,  definitions  appear  at  the  beginning  of  subjects, 
and  principles  are  deduced  from  the  solutions  of  character- 
istic examples.  And  still  later,  noticeably  in  proportion, 
propositions  are  first  enunciated  and  then  logically  proved. 
Thus,  the  pupil  is  led  by  easy  transition  from  the  more 
elementary  forms  of  reasoning  to  pure  mathematical  dem- 
onstration. 

t 83680 


iv  PREFACE, 

In  numerous  instances,  after  deducing  one  or  more  prin- 
ciples, I  have  introduced  selections  of  easy  examples  to  be 
worked  at  sight.  These  are  intended  to  give  opportunity 
for  the  application  of  the  principles  under  which  they 
appear,  and  to  cultivate  in  the  student  a  quick  perception 
of  letter,  exponent,  sign,  and  factor. 

An  unusually  large  number  of  examples  for  written 
work  are  distributed  throughout  the  book.  These  have 
been  selected  with  special  reference  to  the  class  of  pupils 
for  whom  the  work  is  intended.  They  are  arranged  for 
two  readings.  At  the  first  reading  it  is  recommended  that 
all  miscellaneous  examples,  which  are  generally  more  diffi- 
cult than  the  others,  shall  be  omitted.  These,  in  connec- 
tion with  a  review  of  the  definitions  and  principles,  will 
form  a  good  second  reading.  Long,  pointless  examples, 
requiring  much  time  and  labor  in  their  solution,  have  been 
generally  avoided. 

The  rather  extensive  treatment  of  factoring,  and  the 
preparation  provided  for  it  by  the  introduction  of  a  partial 
treatment  of  involution,  a  treatise  on  composition,  and  one 
on  exact  division,  it  is  believed  will  be  commended  by 
teachers  generally.  No  one  can  expect  to  make  much 
progress  in  the  study  of  algebra  who  is  not  somewhat  of 
an  adept  in  factoring. 

The  early  introduction  of  the  equation,  and  the  frequent 
return  to  it,  are  features  so  well  adapted  to  practical  work 
that  comment  upon  their  merits  is  unnecessary. 

The  simplicity  of  the  treatment  of  generalization  and 
specialization,  negative  solutions,  inequalities,  binomial 
surds,  and  limiting  ratios,  is  a  sufficient  excuse  for  their 
introduction  into  an  elementary  treatise  on  algebra.  These 
subjects  may,  however,  be  omitted  where  a  shorter  course 


PREFACE.  y 

is  desirable,  without  doing  violence  to  the  logic  of  other 
parts. 

In  conclusion,  I  desire  to  express  my  deep  obligations  to 
my  wife,  Annie  M.  Sensenig,  whose  experience  as  teacher 
has  been  nearly  coextensive  with  mine,  and  from  whom  I 
have  received  many  practical  helps  and  encouragements  in 
the  preparation  of  this  work. 

I  am  also  greatly  indebted  to  Prof.  A.  J.  Rickoff,  of 
New  York,  for  a  careful  examination  of  the  manuscript 
before  publication,  and  for  many  practical  hints  obtained 
through  his  criticisms. 

David  M.  Senseis'ig. 

Normal  School,  West  Chester,  Pa.,  | 
June  1,  1888.  \ 


CONTENTS. 


INTRODUCTION. 

PAGE 

Literal  quantities — Ideas  and  expression 1 

Kinds  of  literal  quantities 5 

Concrete  examples  involving  literal  quantities     .        .        .        .7 

Positive  and  negative  quantities 10 

Definitions  of  quantities 12 


CHARTER  I. 

INTEGRAL  QUANTITIES. 

Algebraic  addition 13 

Definitions  of  addition 15 

Principles  of  addition  and  applications 15 

Addition  of  similar  monomials 18 

Addition  of  dissimilar  monomials 19 

Addition  of  monomials  with  common  factor        .        .        ,        .20 

Algebraic  subtraction 21 

Definitions  of  subtraction 22 

Principle  of  subtraction  and  applications 23 

Subtraction  of  monomials 25 

Algebraic  multiplication 26 

Definitions  of  multiplication 29 

Multiplication  of  monomial  by  monomial 30 

Multiplication  of  polynomial  by  monomial 31 

Algebraic  division 32 

Definitions  of  division 35 

Division  of  monomial  by  monomial 36 

Division  of  polynomial  by  monomial 37 

Simple  numerical  equations 39 

Definitions  of  simple  equations 41 

Axioms  of  algebra 42 

Concrete  examples  in  simple  equations 44 


viii  CONTENTS, 

PAGE 

Addition  of  polynomials 48 

Subtraction  of  polynomials 49 

Symbols  of  aggregation — Definitions  and  principles       .        .        .50 

Simplification  of  parenthetical  expressions 53 

Inclosing  terms  by  symbols  of  aggregation 53 

Multiplication  by  polynomials 54 

Division  by  polynomials 55 

Simultaneous  equations  of  two  unknown  quantities        .        .        .57 

Definitions  of  simultaneous  equations 58 

Elimination  by  addition  and  subtraction 59 

Concrete  examples  in  simultaneous  equations      .        .        .        .60 

Partial  treatment  of  algebraic  involution — Definitions  and  prin- 
ciples   63 

Involution  of  monomials 64 

Squaring  of  binomials — Principles  and  applications  .  .  .65 
Cubing  of  binomials — Principles  and  applications       .        .        .66 

Composition — Definitions  and  general  principles   '.        .        .        .67 

Special  principles  and  applications 69 

Cross-multiplication — Principle  and  application .        .        .        .71 

Exact  division — Definitions  and  principles 73 

Application  of  principles 76 

Factoring — Definitions  and  principles      .        .        .        .        .        .77 

Factoring  polynomials  with  common  factor  in  terms  .        .        .78 

Factoring  the  difference  of  two  squares 79 

Factoring  the  sum  or  difference  of  equal  odd  powers  .  .  .79 
Factoring  trinomials  that  are  perfect  squares  .  .  .  .80 
Factoring  trinomials  composed  of  binomial  factors  having  a 

common  term 81 

Factoring  trinomials  composed  of  any  binomial  factors  .  .  83 
Factoring  trinomials  composed  of  any  trinomial  factors     .        .    83 

Factoring  polynomials 84 

Miscellaneous  examples  in  factoring 85 

Highest  common  divisor— Definitions  and  principles     .        .        .86 

Highest  common  divisor  of  monomials 87 

Highest  common  divisor  of  polynomials 88 

The  lowest  common  multiple — Definitions 90 

Lowest  common  multiple  of  monomials 91 

Lowest  common  multiple  of  polynomials 93 

Cancellation — Definitions  and  principles 93 

Multiplication  and  division  by  cancellation 94 

Simultaneous  equations  of  three  unknown  quantities — Elimina- 
tion by  addition  and  subtraction 96 

Concrete  examples  in  simultaneous  equations      .        .        .        .97 


CONTENTS,  ix 

CHAPTER  II. 

ALGEBRAIC   FEACTIONS. 

PAQE 

Preliminary  definitions 99 

Reduction  of  fractions — Definition  and  principles  .        .        .        .101 

Ileduction  of  fractions  to  lowest  terms 105 

Reduction  of  mixed  quantities  to  improper  fractions  .        .        .  IOC 
Reduction  of  fractions  to  whole  or  mixed  quantities  .        .        .  107 

Reduction  of  fractions  to  similar  forms 108 

Addition  and  subtraction  of  fractions 110 

Multiplication  and  division  of  fractions 113 

Multiplication  and  division  by  fractions — Definitions  and  prin- 
ciples   115 

Written  examples 117 

Complex  fractions 119 

Involution  of  fractions 120 

Miscellaneous  examples  in  fractions 121 

CHAPTER  III. 

GENERAL   TREATMENT   OF   SIMPLE   EQUATIONS. 

General  definitions 124 

Transformation  of  equations — Definition  and  principles         .        .  125 
Simple  equations  of  one  unknown  quantity — Solution  of  numeri- 
cal equations 126 

Solution  of  literal  equations 129 

Miscellaneous  equations 130 

Concrete  examples 131 

Simple  equations  of  two  unknown  quantities — Definitions  and 

principles 140 

Elimination  by  substitution 141 

Elimination  by  comparison 142 

Solution  of  numerical  equations 143 

Solution  of  literal  equations 146 

Concrete  examples 147 

Simple  equations  of  three  unknown  quantities — Solution  of  ab- 
stract equations 152 

Concrete  examples 155 

Generalization  and  specialization 157 

CHAPTER  IV. 

POWERS   AND   ROOTS. 

Involution  of  binomials — The  binomial  theorem     ....  161 
Involution  of  polynomials — Principles  and  applications         .        .  163 


X  CONTENTS. 

^  PAGE 

Algebraic  evolution — Definitions  and  principles      .        .        .        .165 

Roots  of  numerical  quantities  by  factoring 107 

Roots  of  monomials 168 

Square  root  of  a  polynomial 169 

Square  root  of  numbers 172 

Cube  root  of  polynomials 175 

Cube  root  of  numbers 177 

Higher  roots 180 

Factoring  with  the  aid  of  evolution 180 

CHAPTER  V. 

QUADRATIC   EQUATIONS. 

Quadratic  equations  of  one  unknown  quantity — Pure  quadratics 

— Definitions  and  principles 182 

Solution  of  pure  quadratics  .        .        .        .        .        .        .        .  183 

Affected  quadratics — Definitions  and  principles      ....  185 

Solution  of  numerical  affected  quadratics 185 

Solution  of  literal  affected  quadratics 189 

Equations  in  the  quadratic  form 190 

Solution  of  equations  by  factoring   .        , 192 

Formation  of  quadratic  equations 194 

Formation  of  equations  by  composition 195 

Miscellaneous  equations 197 

Concrete  examples 198 

Quadratic  equations  of  two  unknown  quantities — Definitions        .  201 

Solvable  classes — Illustrations 202 

Solution  of  homogeuQous  equations 207 

Miscellaneous  equations .        .  208 

Concrete  examples 209 

Negative  solutions 211 

CHAPTER  VI. 

EXPONENTS,   RADICALS,  AND   INEQUALITIES. 

Fractional  and  negative  exponents — Principles  and  applications  .  214 

General  principles  of  exponents 219 

Miscellaneous  examples  in  exponents   .        .        .        .        .        .  221 

Radicals — Definitions  and  principles 222 

Reduction  of  radicals — Mixed  to  pure 224 

To  lower  degree     .        . 225 

Rational  to  radical  quantities       .......  226 

To  same  degree 227 

Addition  and  subtraction  of  radicals 228 


COJ^TENTS.  xi 

PAGE 

Multiplication  of  radicals 229 

Division  of  radicals 230 

Involution  of  radicals 231 

Evolution  of  radicals 232 

Rationalization 233 

Imaginary  quantities 234 

Square  roots  of  binomial  surds 236 

Miscellaneous  examples  in  radicals 238 

Radical  equations 242 

Character  of  the  roots  of  equations 245 

Inequalities — Definitions  and  principles 248 

Examples 250 

CHAPTER  VII. 

RATIO,   PROPORTION,  AND  PROGRESSION, 

Ratio — Definitions  and  principles    .......  251 

Examples 253 

Proportion — Definitions 255 

Propositions 256 

Solutions  of  equations  by  proportion 264 

Examples  involving  proportion 265 

Limiting  ratios— Definitions  and  principles 267 

Examples 272 

Arithmetical  progressions — Definitions  and  principles    .        .        .  273 

Examples  involving  arithmetical  progression       ....  274 

Concrete  examples  in  arithmetical  progression     ....  276 

Geometrical  progression — Definitions  and  principles       .        .        .  279 

Examples  in  geometrical  progression 280 

Concrete  examples  in  geometrical  progression      ....  282 

Infinite  series — Definitions  and  principles 285 

Examples  involving  infinite  series 286 

CHAPTER  VIII. 

MISCELLANEOUS  EXAMPLES. 

Abstract  examples 288 

Concrete  examples 293 

General  definitions 298 

Principles  reviewed 300 

APPENDIX 310 


NUMBERS   SYMBOLIZED. 


INTRODUCTION. 
LITERAL   QUANTITIES— IDEAS  AND  EXPRESSION 


EXERCISE    1. 

1.  What  is  the  sum  of  2  units,  3  units,  and  4  units? 

2  tens,  3  tens,  and  4  tens  ?  2  fives,  3  fives,  and  4  fives  ? 
What,  then,  is  the  sum  of  2  times  any  number,  3  times 
that  number,  and  4  times  that  number  ? 

2.  If  we  let  a  stand  for  any  number,  what  will  be  the 
sum  of  2  times  a,  3  times  a,  and  4  times  a  ?  3  times  a, 
4  times  «,  and  6  times  a  ? 

Two  times  a  is  written  2a,  and  is  read  two  a;  three  times  a  is 
written  3a;  etc. 

3.  What  is  the  sum  of  4 «,  5  a,  and  6a?  8  a,  4  a, 
and  7a? 

4.  If  we  let  b  stand  for  any  number,  what  will  be  the 
sum  of  4:b,  3b,  and  2b?    6b,  ^b,  and  Qb? 

In  algebra,  any  letter  may  stand  for  any  number. 

5.  What  is  the  sum  of  3  J,  4i,  and  2by  it  b  stands  for 

3  ?    lib  stands  for  4  ? 

The  symbol  of  addition  is  + ,  read  plus, 

6.  What  is  the  sum  of  2m4-3m-|-5w?  What  when 
m  equals  2  ?    When  m  equals  5  ? 

The  symbol  =  is  read  equals. 


2  ELEMENTARY  ALGEBRA. 

7.  What  is  the  value  of  ^x-\-4:X-\-Qx^    What  when 
a;  ==  3  ?    When  a;  =  6  ? 

8.  bx-\-^x-\-^x=  what  ?    4^  +  5?z  +  ^  =  what  ? 


9.  What  is  the  difference  between  8  tens  and  3  tens? 
8  20's  and  3  20's  ?  8  times  any  number  and  3  times  that 
number?  What,  then,  is  the  difference  between  8a  and 
3a?    8m  and  3w? 

The  symbol  of  subtraction  is  — ,  read  minus. 

10.  What  is  the  value  of  15a;  —  7a;  ?    12y  —  5y? 

11.  What  is  the  value  of  12  a  —  7  a  ?  What  when 
a  =  3  ?    When  a  =  7  ? 

12.  What  is  the  value  of  6a-\-5a—7  a?  What  when 
«  =  5  ?    When  a  =  8  ? 


13.  What  is  the  value  of  a  times  b  when  a  =  3  and 
5  =  4?    When  a  =  6  and  ^>  =  7  ? 

The  symbol  of  multiplication  is  x ,  read  times. 

14.  What  is  the  value  ot  x  X  y  when  x  =  6  and  y  =  S? 
When  a;  =  10  and  y  =  9? 

The  product  of  two  or  more  letters  is  expressed  by  writing  them 
together  without  any  symbol  between  them.  Thus,  a  x  b  =  ab,  and 
a  X  b  X  c  =  abc. 

15.  What  is  the  product  otmXn?pXq?  xXy  Xz? 

16.  What  is  the  product  of  p  X  q  X  r?  What  when 
p  =  2y  q  =  3,  and  r  =  4  ?   When  J9  =  3,  q  =  4:,  and  r  =  5  ? 

17.  What  is  the  value  of  2 a  X  3  J,  when  a=6  and 
^  zz:  7  ?    When  a  =  6  and  Z*  =  3  ? 


18.  What  is  the  quotient  of  x  divided  by  5  when  rr  =  10  ? 
When  a;  =  15  ?    When  ic  =  30  ? 

19.  What  is  the  value  of  a  divided  by  J^when  a  =  15 
and  ^>  =  5  ?    When  a  =  24  and  ^>  =  6  ? 


LITERAL   QUANTITIES.  g 

The  symbol  of  division  is  -f-,  read  divided  hy.    Division  is  also 
expressed  by  writing  the  dividend  over  the  divisor  with  a  line  between 

d 
them.     Thus,  a  divided  by  b  is  written  a-i-b,  or  y . 

20.  What  is  the  value  oi  x-~y  when  x=lb  and  y  =  6? 
When  x  =  (j3  and  y  =  9? 

771 

21.  What  is  the  value  of  —  when  m  =  12  and  7i  =  3? 

n 

When  m  =  18  and  w  =  6  ? 

22.  What  is  the  value  of  —  when  a  =  12,  5  =  0,  and 

c 

c  =  9  ?    When  ff  =  10,  2>  =  7,  and  c  =  5  ? 


23.  What  two  numbers  multiplied  together  will  produce 
10?    15?    21?    da?    5x?    ay?    xz? 

24.  What  three  numbers  multiplied  together  will  pro- 
duce 12  ?    18?    30?    10a?    5ab?    xyz? 

The  numbers  multiplied  together  to  produce  a  given  number  are 
the  factors  of  that  number. 

25.  Name  the  two  factors  of  14.     21.     6  m,     c  d. 

26.  Name  the  three  factors  of  10 2;.     6 ay.    pq r. 


27.  What  number  is  produced  by  using  2  twice  as  a 
factor  ?    Three  times  ?    Four  times  ? 

The  result  obtained  by  using  a  number  two  or  more  times  as  a 
factor  is  a  power  of  the  number.  When  tlie  number  is  used  twice  as  a 
factor  the  result  is  called  the  square  of  the  number.  When  used  three 
times,  the  cube  of  the  number.  When  used  four  times,  the  fourth 
power  of  the  number,  etc.  • 

28.  What  is  the  square  of  3  ?  The  cube  of  4  ?  The 
fourth  power  of  2  ? 

29.  What  is  the  square  of  a  when  a  =  4  ?  The  cube 
of  X  when  a;  =  3  ? 

The  symbol  of  power  is  a  number  called  an  exponent,  written  on 
the  right  hand  above  the  number  whose  power  is  to  be  obtained. 
Thus,  a  squared  is  written  a* ;  a  cubed,  a^ ;  a  fourth  power,  a*,  etc. 


4  ELEMENTARY  ALGEBRA. 

30.  What  is  the  value  of  x^  when  a;  =  2  ?    When  :c  =  3  ? 
When  ^  =  4  ? 

31.  What  is  the  value  of  a?  y^  when  ic  =  2  and  ?/  =  3  ? 
When  a;  =  1  and  ?/  =  4  ? 

32.  What  are  the  factors  of  a^  ?    -^^s  p    ^4  p    ^,2^3  p 


33.  What  is  one  of  the  two  equal  factors  of  4  ?  9  ? 
16?     a^p     ^2p 

34.  What  is  one  of  the  three  equal  factors  of  8  ?    x^Y 

21!  a^? 

One  of  the  equal  factors  of  which  a  number  is  composed  is  a  root 
of  the  number.  One  of  the  two  equal  factors  is  the  square  root ;  one 
of  the  three  equal  factors,  the  cube  root ;  one  of  the  four  equal  factors, 
the  fourth  root,  etc. 

35.  What  is  the  square  root  of  16  ?    25  ?    a^  ?    a^x^  ? 

36.  What  is  the  cube  root  of  27  ?    64  ?    ^^  ?    a^a^? 

The  symbol  of  root  is  a/,  called  the  radical  sign.  A  number  called 
the  index  is  written  in  the  angle  of  the  sign  to  show  the  kind  of  root. 
When  no  index  is  used  the  square  root  is  expressed.  Thus,  y/x  is  the 
square  root  of  x,  and  l/y  is  the  cube  root  of  y. 

37.  What  is  the  value  of  V^  when  x  =  16?  When 
a;  =  49  ? 

38.  What  is  the  value  of  V«  when  a  =  21!  ?  When 
a  =  64  ? 

39.  What  is  the  value  of  V^  ?     VaF  ?     V^? 

40.  What  is  the  value  of  Va^  when  «  =  4  and  x  =  9  ? 
When  a  =  2  and  x  =  8? 

41.  Write  the  square  of  m  ;  the  cube  of  n ;  the  product 
of  m  and  71 ;  tlie  quotient  of  m  and  n ;  the  square  of  m 
divided  hj  the  square  of  n. 

In  algebra,  numbers  expressed  by  figures  only  are  called  numerical 
quantities  ;  and  those  expressed  by  letters  only,  or  by  both  figures  and 
letters,  literal  quantities.  Thus,  24  is  a  numerical  quantity,  and  a,  x^, 
and  3  h  are  literal  quantities. 


LITERAL  QUANTITIES.  6 

Kinds  of  Literal  Quantities. 

EXERCISE    2. 

1.  What  is  the  sum  of  a  and  h  when  a  =  3  and  5  =  5? 
When  a  =  8  and  6  =  9? 

The  sum  of  different  literal  quantities,  when  their  values  are  not 
given,  is  expressed  by  simply  writing  plus  between  them.  Thus,  the 
sum  of  a  and  6  is  a  +  6,  and  of  a,  h,  and  c  is  a  +  b  +  c. 

2.  What  is  the  sum  of  x  and  y?  x,  y,  and  ;?  ?  2  «  and 
35?    4a:,  5y,  and  6;z? 

3.  What  is  the  difference  of  x  and  y  when  a;  =  10  and 
y  =  5  ?    When  ic  =  12  and  2^  =  6  ? 

The  difference  of  different  literal  quantities,  when  their  values  are 
not  given,  is  expressed  by  simply  writing  minus  between  them.  Thus, 
the  difference  of  a  and  h  is  written  a  —  h. 

4.  What  is  the  difference  of  m  and  w  ?  2  a  and  3  5? 
5  7?  and  7  y^  ?    a:^  and  ?/^  ? 

The  different  parts  of  which  a  sum  or  difference  is  composed  are 
called  terms.    Thus,  2 a,  3 6,  and  4 c  are  the  terms  of  2a  +  3&  +  4c. 

5.  Name  the  terms  in  x-\-y  -\-z,  2a-\-ZJ)-\-4:Z, 
X  —  y.     x-\-y  —  z.     X  —  2y-\-3z. 

When  a  quantity  consists  of  only  one  term  it  is  called  a  monomial ; 
when  of  two  or  more  terms,  a  polynomial.  Thus,  3  a  6  is  a  monomial, 
and  2 a  +  3 6  and  4a  —  26  +  c  —  d!  are  polynomials. 

A  polynomial  of  two  terms  is  a  binotnial,  and  one  of  three  terms  a 
trinomial. 

6.  What  are  the  values  of  the  following  binomials, 
when  a  =  Q  and  5  =  3? 

1.  a  +  5        3.  a-\-ab        5.  ab-b-        7.  a^b-ab- 

2.  a-b        4.  a2_^2        g.  a^-b""         8.  2a  +  35 

7.  What  are  the  values  of  the  following  trinomials, 
when  a;  =  10,  y  =  6,  and  z  =  4  ? 

l'^  +  y  +  2^        ^'  X  —  y  —  z  ,  1.  2x  —  y -\-z 

%  x  —  y-\-z        5.  x-\-2y  —  z  8.  2x-\-dy  +  2z 

3.  a;  +  ?^  — ;z        6.  3a;  — 2«/4-2        9.  5a;  — 3?/  +  2^ 


6  ELEMENTARY  ALGEBRA. 

When  terms  have  the  same  letters  affected  by  the  same  exponents 
they  are  similar.    Thus,  3  a^  b\  5  a^  b^,  and  6  a*  b^  are  similar  terms. 

8.  Arrange  the  following  terms  into  groups,  placing 
similar  terms  into  one  group  : 

2ab^  3aH,  4.al)\   ^aH,  ha^W,   ^aH,  1  ah^,  ^a^W, 

The  numerical  factor  in  a  term  is  generally  called  the  coefficient  of 
the  term,  but  any  factor  may  be  taken  as  the  coefficient.  When  no 
numerical  coefficient  is  expressed  the  factor  1  is  understood  to  be  the 
numerical  coefficient. 

9.  Name  the  numerical  coefficients  of  Zax,  ^hc,  Qmn, 
ax,  4:cd,  6x^y^,  a^a^,  2hxy,  m^n. 

Sometimes  terms  have  factors  that  are  alike  and  some  that  are 
unlike ;  then  the  unlike  ones  are  taken  as  the  coefficients  of  the  terms, 
and  the  terms  are  considered  similar  with  respect  to  the  like  terms. 
Thus,  axy,bxy,  and  cxy  sue  similar  with  respect  to  xy.  a,  b,  and  c 
are  the  coefficients. 

10.  With  respect  to  what  letters  are  the  following  terms 
similar,  and  what  are  the  coefficients  of  the  terms  ? 

1.  ax,  hx,  and  ex  3.  2 ax,  3b x,  and  4=  ex 

2.  cxy,  dxy,  and  exy         4.  2my,  3ny,  and  4:sy 

The  symbol  ( ),  called  a  parenthesis,  is  used  to  inclose  two  or  more 
terras  that  are  to  be  taken  together  as  one  factor  or  one  term.  Thus, 
a  +  b  multiplied  by  c  is  written  {a  +  b)c,  and  b  +  c  subtracted  from 
a  is  written  a  —  {b  +  c). 

11.  Find  the  value  of  (4  a  —  2 1))c  when  a  =  3,  b  =  2, 
and  c  =  4. 

If  a  and  b  represent  two  numbers,  what  will  represent 

12.  Their  sum  ?  16.  The  square  of  their  sum  ? 

13.  Their  difference  ?        17.  The  sum  of  their  squares  ? 

14.  Their  product  ?  18.  The  cube  of  their  sum  ? 

15.  Their  quotient  ?  19.  The  sum  of  their  cubes  ? 

20.  The  product  of  their  sum  and  difference  ? 

21.  Their  product  times  their  difference  ? 

22.  The  quotient  of  their  sum  and  difference  ? 


LITERAL  QUANTITIES.  7 

Concrete  Examples  involving  Literal  Quantities. 

EXERCISE    3. 

1.  A  boy  paid  a  cents  for  a  slate  and  h  cents  for  a  book. 
What  did  he  pay  for  both  ? 

Solution. — lie  paid  for  both  the  sum  of  a  cents  and  h  cents,  which 
is  a  +  &  cents. 

2.  I  paid  2a^  cents  for  an  apple  and  ^a  cents  for  an 
orange.     What  was  the  cost  of  both  ? 

3.  A  man  had  5  a  dollars  and  spent  2  a  dollars.     How 
much  money  had  he  left  ? 

4.  Mary  bought  a  lemon  for  3  a  cents  and  an  orange  for 
h  cents.     What  did  she  pay  for  both  ? 

5.  Thomas  rode  Qx  miles  and  then  walked  4ic  miles. 
How  far  did  he  go  in  all  ? 

6.  Mary  had  15  a  quarts  of  berries  and  sold  9  a  quarts. 
How  many  quarts  had  she  remaining  ? 

7.  A  boy  bought  an  apple  for  c  cents  and  handed  over 
a  10-cent  piece.     How  much  change  should  he  receive  ? 

8.  I  bought  a  horse  for  160  and  sold  it  for  y  dollars. 
How  much  did  I  gain  ? 

9.  What  will  be  the  cost  of  3  chairs  at  x  dollars  apiece, 
and  4  tables  at  y  dollars  apiece  ? 

10.  I  bought  m  sheep  at  $6  apiece  and  sold  them  at  19 
apiece.     What  did  I  gain  ? 

11.  If  8  ropes,  each  h  feet  long,  be  cut  from  a  coil  con- 
taining a  yards,  how  many  feet  will  remain  ? 

12.  If  X  acres  of  land  are  worth  $1000,  what  is  the 
value  per  acre  ? 

13.  If  6  horses  are  worth  5  y  dollars,  what  are  10  horses 
worth  at  the  same  price  per  head  ? 

14.  At  h  dollars  a  head,  how  many  horses  will  c  sheep 
at  d  dollars  apiece  buy  ? 


8  ELEMENTARY  ALGEBRA. 

15.  At  m  cents  apiece,  how  many  apples  will  $1  buy  ? 

16.  At  $6  apiece,  how  many  pigs  will  x  dollars  buy  ? 

17.  A  bought  a  farm  of  m  acres  at  n  dollars  an  acre, 
and  sold  it  at  r  dollars  an  acre.     What  was  his  gain  ? 

18.  If  a  bushels  of  wheat  cost  $60,  what  will  x  bushels 
cost  at  the  same  price  ? 

19.  If  a  men  can  do  a  piece  of  work  in  m  days,  in  how 
many  days  can  b  men  do  it  ? 

20.  A  man  bought  a  farm  of  a  acres  at  x  dollars  an  acre, 
and  sold  it  at  y  dollars  an  acre.     How  much  did  he  gain  ? 

21.  At  m  cents  a  pound,  how  many  pounds  of  sugar 
are  worth  as  much  as  c  pounds  of  coffee  at  d  cents  a  pound  ? 

22.  A  is  a  years  old  and  B  is  twice  as  old.  What  will 
be  B's  age  20  years  hence  ?    What  was  it  10  years  ago  ? 

23.  A  and  B  start  from  the  same  place  at  the  same  time 
and  travel  in  the  same  direction.  If  A  travels  m  miles  a 
day  and  B  n  miles  a  day,  how  far  apart  will  they  be  in  c 
days  ? 

24.  What  will  be  the  cost  of  a  rectangular  piece  of  land 
X  rods  long  and  y  rods  wide  at  c  dollars  an  acre  ? 

25.  What  is  the  interest  of  a  dollars  for  t  years  at  r 
per  cent  ? 

26.  A  man  bought  a  horse  for  p  dollars  and  sold  him 
at  a  gain  of  r  per  cent.     What  did  he  receive  for  him  ? 

27.  What  will  it  cost  to  plaster  a  room  a  feet  long,  h 
feet  wide,  and  c  feet  high  at  d  cents  a  square  yard  ? 

28.  A  bought  A:X  bushels  of  clover-seed  at  c  dollars  a 
bushel,  and  sold  one  half  of  it  at  d  dollars  a  bushel  and 
the  rest  at  cost.     What  did  he  gain  ? 

29.  A  miller  mixed  a  bushels  of  corn  worth  m  cents  a 
bushel  with  c  bushels  of  oats  worth  n  cents  a  bushel.  What 
was  the  value  per  bushel  of  the  mixture  ? 


LITERAL  QUANTITIES.  9 

30.  In  what  time  will  p  dollars  at  r  per  cent  amount 
to  a  dollars  ? 

31.  How  many  board  feet  in  a  plank  m  feet  long,  n 
inches  wide,  and  c  inches  thick  ? 

32.  If  A  can  do  a  piece  of  work  in  a  days,  what  part  of 
it  can  he  do  in  c  days  ? 

33.  I  bought  some  goods  at  a  cents  a  yard  and  sold  them 
at  b  cents  a  yard.     What  was  my  gain  or  loss  per  cent  ? 

34.  A  is  a  rods  ahead  of  B,  and  goes  c  rods  while  B 
goes  d  rods.     How  many  rods  must  B  go  to  overtake  A  ? 

35.  If  A  can  go  a  mile  in  a  minutes  and  B  a  mile  in  h 
minutes,  how  much  will  A  gain  on  B  in  one  hour  ?  In  c 
hours  ? 

36.  What  is  the  value  of  a  square  field  x  rods  long  at 
m  dollars  an  acre  ? 

37.  How  much  larger  is  a  rectangular  tract  of  land  x 
rods  long  and  y  rods  wide  than  a  square  tract  z  rods  long  ? 

38.  What  is  the  weight  of  a  cubical  stone  a  feet  long 
if  c  cubic  feet  weigh  a  ton  ? 

39.  A  has  a  garden  m  feet  long  and  n  feet  wide.  What 
would  be  the  Side  of  a  square  garden  of  equal  area  ? 

40.  How  many  cubical  blocks  x  inches  long  are  equiva- 
lent to  one  block  p  feet  long,  q  feet  wide,  and  r  feet  high  ? 

41.  A  rectangular  field  is  a  yards  long  and  h  yards 
wide.     How  far  is  it  across  it  from  corner  to  corner  ? 

42.  A  ladder  c  feet  long  reaches  to  the  top  of  a  tower  a 
feet  high.  How  far  is  the  foot  of  the  ladder  from  the  base 
of  the  tower  ? 

43.  A  bought  a  horse  for  x  dollars  and  sold  him  to  B  at 
a  gain  of  x  per  cent,  who  again  sold  him  to  C  at  a  gain  of 
X  per  cent.     Wfiat  did  B  gain  ? 


10  ELEMENTARY  ALGEBRA. 

Positive  and  Negative  Quantities. 

EXERCISE    4. 

1.  Does  money  gained  in  business  increase  or  diminish 
one's  capital  ?    Money  lost  has  what  effect  ? 

2.  Distance  traveled  in  the  direction  of  one's  destina- 
tion has  what  effect  upon  one's  journey  ?  Distance  trav- 
eled in  the  opposite  direction  has  what  effect  ? 

3.  Power  applied  to  assist  a  moving  cart  has  what  effect 
upon  the  moving  force  of  the  cart  ?  Power  applied  to 
retard  it  has  what  effect  ? 

Quantities  that  have  directly  opposite  tendencies  in  a  mathematical 
calculation  are  called  positive  and  negative. 

Illustrations. — 1.  If  gains  be  considered  positive,  then  losses  will  be 
negative.    If  losses  be  considered  positive,  then  gains  will  be  negative. 

2.  If  past  time  be  considered  positive,  then  future  time  will  be 
negative.  If  future  time  be  considered  positive,  then  past  time  will 
be  negative. 

3.  If  distance  in  any  direction  be  considered  positive,  distance  in 
the  opposite  direction  will  be  negative. 

It  is  customary,  but  not  essential,  to  consider  quantities  that  ex- 
press favorable  conditions  in  an  example  positive,  and  those  that  ex- 
press unfavorable  conditions  negative. 

4.  Tell  which  of  the  following  quantities  are  positive 
and  which  negative  :  John  earns  110,  spends  $8,  finds  $9, 
loses  112,  gives  a  poor  man  15,  receives  a  reward  of  $6. 

5.  Tell  which  of  the  following  quantities  are  positive 
and  which  negative  :  A  man  deposits  150  in  bank,  then 
*' checks  out"  $30,  then  deposits  120,  then  deposits  $40, 
then  *' checks  out"  $50,  then  deposits  $10,  then  "checks 
out"  $12. 

A  quantity  is  marked  positive  by  writing  the  symbol  +  (plus)  be- 
fore it,  and  negative  by  writing  the  symbol  —  (minus)  before  it. 

6.  Write  the  following  quantities  with  their  proper 
signs  :  Thomas  buys  8  sheep,  sells  7,  buys  9,  sells  6,  buys 
5,  kills  10,  buys  12. 


POSITIVE  AND  NEGATIVE  QUANTITIES.  H 

7.  Write  12  positive  units,  3  negative  units,  5  a  positive 
units,  x-\-y  negative  units,  a-\-h  positive  units. 

8.  Write  the  following  quantities  with  their  proper 
signs  :  A  Philadelphian  bound  for  California  travels  west 
2  a  miles  on  Monday,  east  3  a  miles  on  Tuesday,  west  5  a 
miles  on  Wednesday,  west  4  a  miles  on  Thursday,  east  6  a 
miles  on  Friday,  west  7  a  miles  on  Saturday,  and  rests  on 
Sunday. 

9.  If  a  man  walks  10  miles  in  the  direction  of  his  des- 
tination, and  then  walks  5  miles  in  the  opposite  direction, 
what  effect  do  the  last  5  miles  have  upon  the  first  10  ? 

10.  If  one  boy  pulls  at  a  cart  with  a  force  of  20  pounds, 
and  another  holds  back  with  a  force  of  12  pounds,  what 
effect  does  the  12-pound  force  which  the  second  boy  exerts 
have  upon  the  20-pound  force  exerted  by  the  first  boy  ? 

11.  If  a  man  gains  $15  in  one  transaction  and  loses  $25 
in  another,  what  effect  does  the  gain  have  upon  the  loss  ? 

Positive  and  negative  quantities  tend  to  destroy  each  other  when 
combined  in  an  operation,  and  hence  are  said  to  be  opposed  to  each 
other  in  character, 

12.  Which  is  the  more  favorable  condition,  to  be  merely 
penniless  or  to  be  in  debt  $10  ?  To  rest  or  to  go  6  miles 
in  the  opposite  direction  from  one's  destination  ?  To  be 
idle  or  to  lose  $20  in  business  ? 

A  negative  quantity  is  sometimes  regarded  as  less  than  zero. 

13.  Which  is  the  more  favorable  condition,  to  owe  $5  or 
to  owe  $10  ?  To  lose  10  sheep  or  to  lose  20  sheep  ?  To 
go  8  miles  or  15  miles  in  a  wrong  direction  ? 

Of  two  negative  quantities,  that  is  considered  the  greater  which 
has  the  less  number  of  units. 

14.  One  boy  helps  a  cart  along  with  a  force  of  12  pounds 
and  another  retards  it  with  a  force  of  8  pounds.  Write 
the  combined  effect  of  these  forces  upon  that  of  the  cart. 

15.  How  many  and  what  kind  of  units  are  there  in  +  7  ? 
-6?    -fa?     -^>?    +3a?     -2Z^?    -f«^?     -//? 


12  ELEMENTARY  ALGEBRA. 

16.  A  miller  bought  80  bushels  of  oats  and  sold  95 
bushels  in  one  day.  Write  the  combined  effect  of  these 
transactions  upon  the  amount  of  oats  on  hand. 

17.  A  earns  a  dollars  and  spends  h  dollars.  Write  the 
combined  effect  of  these  transactions  upon  his  finances 

1.  When  a  is  greater  than  h. 

2.  When  a  is  less  than  h, 

18.  A  man  earned  x  dollars  one  day  and  y  dollars  an- 
other. Write  the  combined  effect  of  the  two  days'  wages 
upon  his  finances. 

19.  A  man  spent  a  dollars  at  one  time  and  J  dollars  at 
another  time.  Write  the  combined  effect  of  these  trans- 
actions upon  his  finances. 

20.  A  land-holder  buys  a  tract  of  land  a  rods  long  and 
h  rods  wide.  Write  the  effect  of  this  transaction  upon  the 
amount  of  land  he  owns. 

Definitions. 

1.  A  Unit  is  a  single  thing. 

2.  One  or  more  units  of  a  kind  is  a  Number. 

3.  A  definite  number  of  units  is  a  Specific  Quantity; 
as,  seven  birds. 

4.  An  indefinite  number  of  units  is  a  General  Quan- 
tity; as,  a  jioch  of  birds. 

5.  A  number  expressed  by  figures  only  is  a  Numerical 
Quantity;  as,  125. 

6.  A  number  expressed  by  letters,  or  figures  and  letters, 
is  a  Literal  Quantity;  as,  x  and  6x. 

7.  Numbers  opposed  to  each  other  in  character  are  dis- 
tinguished by  the  symbols  +  (plus)  and  —  (minus),  and 
are  called  Positive  and  Negative  Quantities. 

Note.— For  complete  definitions,  see  pages  298  and  299. 


CHAPTER   I. 
INTEGRAL    QUAJ^TITIES, 


Algebraic  Addition. 

EXERCISE    B. 

1.  A  man  earned  15  one  day,  $4  the  next,  and  $7  the 
next.  What  was  the  combined  effect  of  these  earnings 
upon  his  finances  ? 

Form. 
Solution. — Since  earnings  increase  his  _\      ^k 

money,  we  mark  each  earning  positive.  •" 

The  whole  increase  is  evidently  the  sum  ~r      4 

of  $5,  $4,  and  |7,  which  is  $16,  which  +      '^ 

we  mark  positive.  -|-  $16 

2.  One  boy  helps  a  cart  along  with  a  force  of  16  pounds, 
another  with  a  force  of  20  pounds,  and  another  with  a  force 
of  25  pounds.  What  is  the  combined  effect  of  these  forces 
upon  that  of  the  cart  ? 

3.  A  miller  sold  5  bushels  of  oats  to  one  man,  6  bushels 
to  another,  and  9  bushels  to  another.  What  was  the  com- 
bined effect  of  these  transactions  upon  the  amount  of  oats 
on  hand  ? 

ForiDi 
Solution. — Since  oats  sold  diminishes  p,  , 

the  amount  on  hand,  we  mark  each  quan- 
tity  negative.    The  whole  decrease  is  evi-  " 

dently  the  sum  of  5  bu.,  G  bu.,  and  9  bu.,  —     9    *^ 

or  20  bu.,  which  we  mark  negative.  —  20  bu. 

4.  A  man  sold  10  cows  one  day,  15  the  next,  and  20  the 
next.  What  was  the  combined  effect  of  these  transactions 
upon  the  number  in  his  herd  ? 


+    15 

-    $3 

+      7 

-      6 

+      8 

-      9 

+  $20 

-$18 

-    18 

+   $2 

14  ELEMENTARY  ALGEBRA, 

5.  A  man  earns  $5,  then  spends  $3,  then  earns  $7,  then 
spends  $6,  then  earns  $8,  then  spends  $9.  What  is  the 
combined  effect  of  these  transactions  upon  his  finances  ? 

Solution. — We  mark  all  incomes  posi-  ponn. 

tive,  and  all  outlays  negative.  The  sum 
of  the  incomes  is  $20,  which  we  mark 
positive.  The  sum  of  the  outlays  is  $18, 
which  we  mark  negative.  Now,  an  outlay 
of  $18  will  destroy  an  income  of  $18,  or 
—  $18  will  destroy  +  $18,  and  there  will 
remain  an  income  of  $2,  which  we  mark 
positive. 

Eemark. — A  negative  quantity  will  destroy  a  positive  quantity  of 
the  same  number  of  units  when  combined  with  it. 

6.  Six  men  push  at  a  moving  car.  A  pushes  forward 
80  pounds,  B  backward  90  pounds,  0  forward  100  pounds, 
D  backward  95  pounds,  E  backward  110  pounds,  and  F 
forward  85  pounds.  What  is  the  combined  effect  of  these 
forces  upon  that  of  the  car  ? 

7.  A  drover  adds  2  a  sheep  to  his  flock,  then  sells  3  a, 
then  buys  4  a,  then  sells  3  a,  then  buys  5  a,  then  sells  3  a, 
then  buys  6  a.  What  is  the  combined  effect  of  these  trans- 
actions upon  the  number  in  his  flock  ? 

Combining  algebraic  quantities  is  called  adding  them. 

8.  Find  the  sum  of  +  3  «,  —  4:a,  -{-Qa,  —6  a,  —  3  a, 
and  -\-2a. 

Solution. — The  sum  of  the  positive 
quantities  is  +  11a,  and  the  sum  of  the 
negative  quantities  is  —  12  a.  If  we  com- 
bine 11  a  positive  units  with  12  a  nega- 
tive units,  they  will  destroy  11a  negative 
units,  and  a  negative  units,  or  —  a,  will 
remain. 

Eemark. — Equal  positive  and  nega- 
tive quantities  may  be  omitted  in  addi-  — 
tion,  since  they  destroy  each  other. 

9.  Find  the  sum  of  -f  2,  -  3,  -f  4,  -  5,  and  +  7. 


] 

Form. 

-f    da 

-    4a 

-1-    6a 

-    5a 

-f    2a 

-    da 

+  11  a 

-12  a 

+  11  a 

ALGEBRAIC  ADDITION,  15 

10.  Find  the  sum  of  +  2^>  +  3  o^,  —  4  «,  —  5  a,  +  7  a, 
—  6  a,  and  +  3  fl. 

When  no  sign  is  written  before  an  algebraic  quantity,  +  is  under- 
stood. 

11.  Find  the  sum  of  3  a;,  —  4  a;,  7  a;,  --5  a;,  and  3  x. 

12.  Find  the  value  of  +2m  +  (+3?^)  +  (-2^0  + 

Definitions. 

8.  The  result  obtained  by  combining  two  or  more 
quantities  without  regard  to  their  character  as  positive  or 
negative,  is  the  Arithmetical  Sum  of  the  quantities. 

9.  The  result  obtained  by  combining  two  or  more 
quantities  with  regard  to  their  character  as  positive  or 
negative,  is  the  Algebraic  Sum  of  the  quantities. 

niostration. — If  a  man  goes  10  miles  in  the  direction  of 
his  destination  and  4  miles  in  the  opposite  direction,  the 
entire  distance  traveled,  the  arithmetical  sum,  is  14  miles  ; 
but  the  distance  he  advanced  on  his  journey,  the  algebraic 
sum,  is  only  6  miles. 

10.  The  process  of  finding  the  algebraic  sum  of  two  or 
more  quantities  is  Algebraic  Addition. 

Principles  and  Applications. 

1,  Find  the  sum  oi  -\-'Za,  +  3  a,  and  +  4a ;  also  the 
sum  of  —  2  a,  —  3  a,  and  —  4  a. 

Solutioii. — 1.  The  sum  of  2  a  positive 
units,  3  a  positive  units,  and  4  a  positive  Forms. 

units    is    evidently    9  a    positive    units. 

Therefore,  the  sum  of  -h  2  a,  -I-  3  a,  and         +  2  «  "~  f  ^ 

+  4ais+9a.  4-3a 

2.  The  sura  of  2  a  negative  units,  3  a         -]-  4  a 
negative  units,  and  4 a  negative  units  is  19^ 
9  a  negative  units.    Therefore,  the  sum 
of  —  2  a,  —  3  a,  and  —  4  a  is  —  9  a. 


le  ELEMENTAR Y  ALGEBRA. 

Therefore, 

Principle  1, — The  algebraic  sum  of  two  or  more  similar 
terms  with  like  signs  equals  their  arithmetical  sum  with 
the  same  sign. 

SIGHT     EXERCISES. 

Name  at  sight  the  sum  of  the  following  quantities  : 

1.  2.  3.  4.  6.  6. 

-\-2a        -5x        -6y         -]-dz        + 11  a^        -9ab 
-j-Sa        -7x        -8y        -\-Sz        +    7  a^        -6ab 


7. 

8. 

9. 

10. 

11. 

12. 

+  2«2 

-6x^ 

-4.ah 

+    ^b 

-bax^ 

-]-6bx 

+  3  a' 

-5a^ 

-Qab 

+    8& 

-lax^ 

+  8bx 

+  5a2 

-7x' 

-lab 

+  10^> 

-9ax^ 

+  7bx 

13.  +3a  +  (+4a)  +  {+6a) 

14.  -6ic  +  {-5x)  +  {-2x) 

15.  -\-6x'-{-  {-{-6x')  +  i+Sa^) 

16.  -  5  m^  +  (-  7  m^)  +  (-  3  m^) 


2.  Find  the  sum  of  -{-6a  and  —2a;  also  the  sum  of 
—  6a  and  +  ^ ^• 

Solution. — 1.  If  2  a  negative  units  be  Forms. 

combined  with  5  a  positive  units,  they         _|_  5  //  5  a 

will  destroy  2 a  positive  units,  and  da  ^ 

positive  units  will  remain.     Therefore,  ZL, Jl 

the  sum  of  +  5  a  and  — 2a  is  +3  a.  -\- 3  a  -^  3  a 

2.  If  2  a  positive  units  be  combined 
with  5  a  negative  units,  they  will  destroy  2  a  negative  units,  and  3  a 
negative  units  will  remain.    Therefore,  the  sum  of  —  5  a  and  +  2  a 
is  —3  a. 

Therefore, 

I*rin,  2, — The  algebraic  sum  of  two  similar  terms  with 
unlike  signs  equals  their  arithmetical  difference  with  the 
sign  of  the  greater. 


ALGEBRAIC  ADDITION,  17 

SIGHT     EXERCISES. 

Name  the  sum  of  the  following  quantities  : 

1.                  2.                  3.                 4.                  6.  6. 

+  %a         +6a        -5a        -Ix        ~9a^  +3aJ 

-5a         -3a        +6«        -^2x        -{-7a^  -Sab 

7.  8.  9.  10.  11.  12. 

-23^     -bxy     -lOa^    +122:=^     -    bm7i     -\-3{a-\-b) 
+  8ar^    ±Sxy     +10a^     -    7a:^    -j-nmn     -6(a  +  b) 

13.  -5arH- (+2a:2)  ^g,  _|_a;2^3  ^  (_  ^^3) 

1^  +7xy  -{-  (-3xy)  16.  -  3/^^  _|.  (_|_6y  ^2) 


5.  Find  the  sum  of  +  ^>  +  ^>  and  —  c. 

Solution. — If  b  positive  units  be  added 

to  a  positive  units,  the  sum  will  he  a  +  b  •^™™' 

positive  units ;  if  now  to  «  +  &  positive  -J-  a 

units  c  negative  units  be  added,  they  will  _1_  j^ 

destroy  c  positive  units,  and  a  +  b  —  c  __  ^ 
positive  units  will  remain.     Therefore, 


the  sum  of  +  a,  +  b,  and  _  c  is  +  (a  +  -f-  {a -[^  0  —  c)  Or 

b  —  c),  or  simply  a  +  b  —  c,  the  positive  a  -\-  0       C 

sign  being  understood. 

Bemark. — If  c  were  numerically  greater  than  a  +  b,  the  sum  would 
be  c  —  {a  +  b)  negative  units,  which,  as  will  be  learned  in  subtraction, 
would  still  be  a  +  &  —  c. 

Therefore, 

Prin,  3, — The  algebraic  sum  of  two  or  more  dissimilar 
terms  equals  a  polynomial  composed  of  those  terms, 

SIGHT     EXERCISES. 

Name  the  sum  of  the  following  quantities  : 

1.  a,  +3^,  and  —2c  4.  2  a; +(— 3  «/)  +  (— 42;) 

2.  2x,  -^y,  and  +3^  5.  1  z^ -\- (-\- 2  z)  +  (- b) 

3.  bz\  -7/,  and  -(Jar  6.  S p^ -\- (^  ^ q^) -^  {-1  r") 


18  ELEMENTARY  ALGEBRA. 

Problem  1.    To  add  similar  monomials. 

Illustration.— Find  the  sum  of  +  3  «,    —  4  «,   +  6  «, 

—  5  «,  —3  a,  and  -\-2a. 

Solution. — The  sum  of   the  positive  Form, 

quantities  is  +  11  a  [P.  1],  and  the  sum 
of  the  negative  quantities  is  —  2  a  [P.  1]. 
Now,  the  sum  of  +  11  a  and  — 12  a  is 

-  a  [P.  2]. 

Eemark.— If  preferred,  explain  as  on 
page  14. 

Suggestion  to  Teacher.— Require  pu-  —  a  . 

pils  to  recite  principles  whenever  refer- 
ence is  made  to  them  in  solutions.    Do  not  demand  the  numbers  of 
principles. 

EXERCISE    6. 

Find  the  sum  of  the  following  columns  : 


+    Sa 

-    4.a 

4-    6a 

—    6a 

+    2a 

-    da 

+  11  a 

—  12  a 

+  11  a 

1. 

2. 

3. 

4. 

-\-2a 

-6x 

+  2xy 

-Sab 

+  5a 

-Ix 

-"Ixy 

+  4:ab 

+  6« 

-%x 

-\-bxy 

-6ab 

+  7a. 

-4.x 

-^xy 

+    ab 

6. 

6= 

7. 

8. 

Sax 

-6x^y 

—  6mn 

-Spq 

—  6ax 

^7?y 

—  Omn 

7pq 

—  4,ax 

-%x^y 

Imn 

Spq 

-\-Qax 

-^7?y 

dmn 

-9pq 

+  2ax 

10. 

—  Smn 

-    pq 

9. 

11. 

12. 

d{a  +  b) 

—  3  (m  —  n) 

H^  +  f) 

-  Hx+yy 

-^{a  +  b) 

7(m-n) 

S{x^  +  f) 

-    nx  +  yf 

Q{a  +  b) 

—  6{m  —  71) 

-n^+/) 

+    e(x  +  yr 

2{a-{-b) 

8  (m  —  n) 

-6{ar-j-f) 

+   S{x  +  yY 

-.6{a  +  h) 

—  4:(m  —  n) 

-3(^  +  /) 

- 10  (x  +  yY 

ALGEBRAIC  ADDITION,  19 

13.  Add  4  a"  b\  -lar  W,  8  a^  W,  -  6  «« W,  and  12  a^  b\ 

14.  Add  —6xyZf  —9xyz,  Ixyz,  —^:xyz,  and  ^xyz. 

16.  Add   3  (a  —  m),  —  7  (a  —  wi),  6  («  —  ?^i)>  and 

—  8  («  —  m). 

Note. — When  quantities  are  written  in  succession,  separated  by 
positive  and  negative  signs,  their  sum  is  intended.  Thus,  3  a  -  5  a  + 
7a  — 4a  =  (3a)  +  (-5a)  +  (+  7a)  +  (-4a). 

16.  Collect  into  one  quantity  7a  —  4a  +  5a  —  6a4-'^^ 

—  a. 

17.  Collect  9J-7*+6J-95  +  85-i-2J  +  35 
-lb, 

18.  Collect  -^ab-\-'iab-10ab-{-dab-6ab^ 
7  ab  —  Sab, 

19.  Collect    9  m^n^  —  S  m^  n^  —  12  m^  7i^  +  7  m^  n^  — 

20.  Collect  3  (a  +  ^)  -  5  (a  +  ^)  +  'J'  («  +  *)  -  6  («  +  J) 
+  4(a  +  Z')-8(a  +  J)  +  6(a  +  ^)-5(a  +  &). 

Problem  2.    To  add  dissimilar  monomials. 

Illustration.— Find  the  sum  of 
3  a,  —  5  &,  and  -f-  2  c. 

Solution. — Since  the  algebraic  sura  of 
dissimilar  terms  equals  a  polynomial 
composed  of  those  terms  [P.  3],  the  sum 
of  3  a,  —  5  6,  and  +2cis3a  —  5&H-2c. 

EXERCISE    7. 

Add  the  following  columns  : 
1.  2.  3. 

X  2x  5a 

y         -3y  -3b 

z  4:Z  —2c 


Form. 

3a 

-5b 

+  2c 

da 

-5b  +  2c 

6.  Add  7  a  ^,  —  4  c  ^,  5ac, 


4. 

5. 

—  5m 

-ab 

■i-6n 

■\-cd 

+  7r 

-\-4:h 

6bd,  and  4 

a  771, 

20 


ELEMENTARY  ALGEBRA. 


7.  Add  —xy,  -\-ltjz,  —  4:Xz,  -\-9my,  and  —  Gnx. 

8.  Collect    —4:al)  -{-  7xy  —  Sab  —  2xy  -{-  4:ab    and 
-xy. 

Suggestion. — Collect  first  the  similar  quantities,  then  combine  the 
dissimilar  sums. 

9.  Collect  6a-\-4:b  —  3a-\-2I)  —  5a-^l!b-\-6a  —  5h. 

10.  Collect  6x-{-4:y  —  dz  +  '7z  —  4:X-\-6y  —  8y  +  10z. 

11.  Collect   3m^  -\-  4:71^  —  5mn-\-7 771^  —  n^-\-6mn  — 
4  m^  —  6  71^. 

12.  Collect  6ax  —  4:by-^'7ax  —  Sax-\-4cby-{-6ax  — 
15  a  X. 


Form. 

a 

xy 

I 

xy 

—  c 

xy 

(a-{-b  —  c)xy 


Problem  3.    To  add  monomials  having  a  common  factor. 

Illustration. — Find  the  sum  of 
axy,  bxy,  and  —cxy. 

Solution. — The  common  factor  is  xy. 
a  times  xy  plus  h  times  xy  is  {a  +  b) 
times  X  y ;  and  {a  -\-  h)  times  x  y  added 
to  —  c  times  xy  is  {a  +  h  —  c)  times  x y 
[P.  3]. 

EXERCISE    8. 

Add  the  following  columns  : 

1.  2.  3. 

ax  ayz  2ay 

hx  —dyz  Siy 

c  X  -\-myz  —  4:cy 


4. 

2xz 

—  axz 

—  hxz 


5.  Add  2 axy,  Zhxy,  ^cxy,  and  —  dxy, 

6.  Collect  anl)  —  amh-\-aph  —  aqh-^ari, 

7.  Collect  axy  —  l)xy-\-cxy  —  2axy-\-3hxy  —  4:Cxy, 

8.  Collect  SaiTny  —  4:hcmy  -\-  Gcdmy  —  badmy, 

9.  Collect  a(c-^d)  -\-l{c-{-  d). 

XO.  Collect  a{x-\-y-Yz)  -\-b{x-\-y-\-z)  —  c{x-\-y-\-z). 


ALGEBRAIC  SUBTRACTION.  21 

Algebraic  Subtraction. 

EXERCISE    9. 

1.  A  gain  of  how  many  dollars  must  be  added  to  a  gain 
of  3  dollars  to  make  a  gain  of  7  dollars  ?  What  then  must 
be  added  to  +  $3  to  make  -j-  $7  ? 

2.  A  loss  of  how  many  dollars  must  be  added  to  a  loss 
of  3  dollars  to  make  a  loss  of  7  dollars  ?  What  then  must 
be  added  to  —  $3  to  make  —  $7  ? 

3.  A  loss  of  how  many  dollars  must  be  added  to  a  gain 
of  7  dollars  to  make  a  gain  of  only  3  dollars  ?  What  then 
must  be  added  to  +  ^7  to  make  +  ^3  ? 

4.  A  gain  of  how  many  dollars  must  be  added  to  a  loss 
of  7  dollars  to  make  a  loss  of  only  3  dollars  ?  What  then 
must  be  added  to  —  $7  to  make  —  $3  ? 

5.  A  gain  of  how  many  dollars  must  be  added  to  a  loss 
of  3  dollars  to  make  a  gain  of  7  dollars  ?  What  then  must 
be  added  to  —  $3  to  make  +  $7  ? 

6.  A  loss  of  how  many  dollars  must  be  added  to  a  gain 
of  7  dollars  to  make  a  loss  of  3  dollars  ?  What  then  must 
be  added  to  -f  $7  to  make  —  $3  ? 

7.  A  loss  of  how  many  dollars  must  be  added  to  a  gain 
of  3  dollars  to  make  a  loss  of  7  dollars  ?  What  then  must 
be  added  to  +  $3  to  make  —  $7  ? 

8.  A  gain  of  how  many  dollars  must  be  added  to  a  loss 
of  7  dollars  to  make  a  gain  of  3  dollars  ?  What  then  must 
be  added  to  —  $7  to  make  +  $3  ? 

9.  A  loss  of  how  many  dollars  must  be  added  to  a  gain 
of  5  dollars  to  make  neither  a  gain  nor  a  loss?  What  then 
must  be  added  to  +  $5  to  make  0  ? 

10.  A  gain  of  how  many  dollars  must  be  added  to  a  loss 
of  5  dollars  to  make  neither  a  gain  nor  a  loss  ?  What  then 
must  be  added  to  —  $5  to  make  0  ? 

The  quantity  that  must  be  added  to  one  of  two  given  quantities  to 
make  the  other  is  the  difference  of  the  quantities.    The  process  of 


f 


^      OF  THE 
!  !  M  1  \/ 


ERSiTY  1 


22  ELEMENTARY  ALGEBRA. 

finding  the  difference  is  subtraction.  The  quantity  formed  of  the 
difference  and  one  of  the  given  quantities  is  the  minuend.  The  quan- 
tity added  to  the  difference  to  form  the  minuend  is  the  subtrahend. 

11.  What  must  be  added  to  +  3  a  to  make  +  7  «  ?  What 
then  is  the  difference  of  +  7  a  and  +  3  a  ?  W^hich  quan- 
tity is  the  minuend,  and  which  the  subtrahend  ? 

12.  What  must  be  added  to  —  3  «  to  make  —  7  «  ?  What 
then  is  the  difference  between  —  7  «  and  —3a?  Which 
quantity  is  the  minuend,  and  which  the  subtrahend  ? 

13.  What  must  be  added  to  -\-^a  to  make  0  ?  What 
then  is  the  difference  between  0  and  +  4  a  ?  Which  quan- 
tity is  the  minuend,  and  which  the  subtrahend  ? 

14.  What  must  be  added  to  —  5  a  to  make  0  ?  What 
then  is  the  difference  between  0  and  —5a?  Which  quan- 
tity is  the  minuend,  and  which  the  subtrahend  ? 

Definitions. 

11.  The  Difference  of  two  quantities  is  such  a  quantity 
as  added  to  one  of  them  will  produce  the  other. 

12.  The  difference  of  two  quantities  without  regard  to 
their  character  as  positive  or  negative  is  their  Arithmetical 
Difference, 

13.  The  difference  of  two  quantities  when  regard  is  had 
to  their  character  as  positive  or  negative  is  their  Algebraic 
Difference. 

Ulustration. — The  difference  between  traveling  7  miles 
and  4  miles,  irrespective  of  direction,  is  3  miles.  This  is 
the  arithmetical  difference.  But  the  difference  made  in 
one's  journey  between  traveling  7  miles  in  the  direction  of 
one's  destination  and  4  miles  in  the  opposite  direction,  is 
an  increase  of  11  miles.     This  is  the  algebraic  difference. 

14.  The  process  of  finding  the  algebraic  difference  of 
two  quantities  is  Algebraic  Subtraction. 


ALGEBRAIC  SUBTRACTION,  23 

15.  The  general  problem  of  algebraic  subtraction  is : 
''Given  the  algebraic  sum  of  two  quantities  and  one  of 
them,  to  find  the  other,'* 

Principle  and  Applications. 

16.  Since  the  difference  of  two  quantities  is  such  a  quan- 
tity as  added  to  the  subtrahend  will  produce  the  minuend, 
it  may  readily  be  found  in  three  steps,  as  follows  : 

1.  Find  what  quantity  added  to  the  subtrahend  will 
produce  zero.  This  is  evidently  the  subtrahend  with  the 
sign  changed. 

2.  Find  what  quantity  added  to  ze7'0  will  produce  the 
minuend.     This  is  evidently  the  minuend. 

3.  The  sum  of  the  two  quantities  thus  added  is  evi- 
dently the  difference.     Therefore, 

JPWn.  4,  —  The  algebraic  difference  of  two  quantities 
equals  the  algebraic  sum  obtained  by  adding  to  the  minuend 
the  subtrahend  with  the  sign  changed. 

Illustration.— Find  the  difference  oi  -\-^a  and  H-8«; 
that  is,  find  what  quantity  added  to  +  8  a  will  produce 

Solution. — 1.  If  we  add  —  8  a  to  +  8  a,  we  will  have  zero, 

2.  If  we  add  +  5  a  to  zero,  we  will  have  +  5  a. 

3.  Therefore,  if  we  add  the  sum  of  —  8  a  and  +  5  a,  or  —  3  a,  to 
+  8  a,  we  will  have  +  5  a.    Hence, 

Difference  of  S       „      v  =  sum  of  ■<       !-,      ?■ 
{   +8a  )  (  —Sa  \ 

—  3a 

Exercise. — Prove  as  in  the  illustration  the  truth  of  the 
following  examples  : 

1.  2.  3.                   4.                   6. 

Minuend,      -\-Sa  — 8a  —5a  -\-   Sa  —   Sa 

Subtrahend,  +5a  —5a  —Sa  —3a  -^   Sa 

Difference,    +3a  —3a  -\-3a  —11  a  —11a 


24  ELEMENTARY  ALGEBRA. 

17.  The  principle  of  algebraic  subtraction  may  also  be 
illustrated  as  follows  : 

1.  Arrange  positive  and  negative  numbers  as  in  the 
following  scale  : 

-10  -9  -8  -7  -6  -5-4  -3  -2  -1      0    +1   +2  +3   +4   +5   +6  +7  +8   +9  +lo 

"1    I     1    \    1    1     \    \    1     I     \    \    I     I     i     I     1)1    \    T" 

2.  Consider  the  difference  of  two  numbers,  the  number 
of  units  passed  over  in  going  on  the  scale  from  one  of  them 
to  the  other. 

3.  Consider  units  passed  over  in  going  from  left  to 
right  positive,  and  from  right  to  left  negative, 

4.  To  find  the  difference,  pass  from  the  subtrahend  to 
zerOy  then  from  zero  to  the  minuend,  and  show  that  the 
algebraic  sum  of  these  distances  equals  the  number  of 
units  that  must  be  passed  over  in  going  directly  from  the 
subtrahend  to  the  minuend. 

niustration. — 1.  Find  the  difference  of  +  3  and  +  8. 

Solution.— From  +8  to  0  is  —  8,  and  from  0  to  +  3  is  +  3 ;  hence, 
from  +  8  to  +  3  would  seem  to  be  the  sum  of  —  8  and  +  3,  or  —  5. 
This  we  see  on  the  scale  is  true. 

2.  Find  the  difference  of  —  3  and  +  8. 

Solution.— From  +  8  to  0  is  -  8,  and  from  0  to  —  3  is  —  3 ;  hence, 
from  +  8  to  —  3  would  seem  to  be  the  sum  of  —  8  and  —  3,  or  —  11. 
This  we  see  on  the  scale  is  true. 

SIGHT     EXERCISE. 

Name  the  difference  of  the  following  quantities  : 

1.  2.  3.  4.  5.  6. 

+  5a  +3a  -ba  -3a         +6a  -3a 

+  3«  +5«  —Sa  —5a  —3a  +5a 


7. 

8. 

9. 

10. 

11. 

12. 

—  5a 

+  da 

-xy 

-^x^ 

-9xy 

-9a;3 

-f  3a 

-5a 

-\-3xy 

-\-9a? 

—  7  a:?/ 

-{-9x^ 

ALGEBRAIC  SUBTRACTION,  25 

13.  7Z>-(+3J)  17.  -42;2-(_2z2) 

14.  —Gx—{'-%x)  18.  —Qxy  —  {-\-lxy) 
16.  8ar^--(-9  2;2)  ^g    _2a;-(+8a;) 
16.  3  3/2- (+6  y2)  20.  +5a;-(-8a;) 

21.  -5.^"^ -(-10 3^3) 

Problem  1.    To  subtract  monomials. 

Ulnstration. — 

1.  Find  the  difference  of  -\-Zah  and  —  5ah, 
Solution : 

DifEcrence  of  )  ^^  (  =sum  of  {  +  ^«*  [  [P.  4]  =  +  Bab. 

Formt 

2.  From  —  Sa      Solution :  Minuend  =  —3a 
take       —  2  b       Subtrahend  with  sign  changed  =            -{-2  b 

2b  — 3a      Difference  [P.  4]  =     2b  — 3a 


3.  From    axy      Solution:  Minuend  =  a 

take    —  bxy      Subtrahend  with  sign  changed  =       -}-  b 


xy 
xji 


(a'^b)xy      Difference  [P.  4]  =  {a-\-b)xy 

EXERCISE     lO. 

I.  From  +  7  fl  take  -\-3a         2.  From  +  6  a  take  +  9  a 

3.  From  —  9 a         4.  From  —3x        5.  From  -\-lb 
take  —ba  take  —%x  take  —  6  5 

6.  From  +3^        7.  From  —^ab      8.  From  —    %xy 
take  -  11  J  take  +  7  a  6  take  -\-l2xy 

9.  From  +   5a      10.  From  -\-b  11.  From  —xy 

take  —  12  d  take  —  a  take  —  3  wi?t 

12.  Find  the  value  of  3a^x^-  (4  a^'x') 

13.  Find  the  value  of  —3m^n^—{—2m^  n^) 

14.  7a,'«/--  (~  6^2/)  =  what  ? 

16.  3a:«/-(+7ww)=  ?  17.  4-c2-(- J«)  =  ? 

16.  —  ?ri2  ,^  _  (__  (5  ^  ^)  _  p        18.  —  7V  —  (—  m^)  =  ? 


26  ELEMENTARY  ALGEBRA. 

19.  From  3(«  +  .'r)  take  4:{a-\-x) 

20.  From  5  {x^  —  y'^)  take  —  4  (a;^  —  ?/^) 

21.  From  8  {x  —  yf  take  —  8  (a;  —  yf 

22.  From  o^?/^  take  —  hy"^         24.  From  ma;  take  n^x 

23.  -ca:2-(-^a;2)^  p  25.  2aa;- (+3  5a;)  =  ? 


Algebraic  Multiplication. 

Principles  of  Signs. 

EXERCISE     11. 

1.  Five  times  4  a  positive  units  are  how  many  positive 
units  ?    Then,  5  (+  4  «)  =  what  ? 

2.  Five  times  ^a  negative  units  are  how  many  negative 
units  ?    Then,  5  (-  4  a)  =  what  ? 

What  is  the  value  of 

3.  3(+5«)?  5.  3(+6rr)?  7.  8(+5y)? 

4.  4(-2a;)?  6.  5(-7«/)?  8.  6  (-3^)? 


9.  What  is  the  meaning  of  the  expression  a:  +  3  (+  2  J)  ? 
Solution :  cc  +  3(+  2  6)  denotes  that  3  times  2&  positive  units  are 

to  be  added  to  x. 

10.  What  is  the  meaning  of  a;  -|-  3  (—  2  5)  ? 

11.  What  is  the  meaning  of  a;  -  3  (+  2  J)  ? 

12.  What  is  the  meaning  of  a:  —  3(— 2Z>)? 


13.  What  is  the  value  of  x -\- ^  {-{•  2  h)  ? 
Solution :  3(+2&)=  +  6&;  hence, 

cc  +  3  (+  2  &)  =  a;  +  (+  6 ^')  =  a;  +  6 &  [P.  3]. 

14.  Since  a;  +  3  (+  2  J)  =  a;  +  6  ^  [Ex.  8],  what  is  the 
value  of  +  3  (+  2  Z*)  ?  Then  a  positive  quantity  multiplied 
by  a  positive  quantity  will  give  what  kind  of  quantity  ? 


ALGEBRAIC  MULTIPLICATION,  27 

15.  What  is  the  value  of  a;  —  3  (-  2  J)  ? 
Solution :  3  (-  2  6)  =  —  6  6 ;  hence, 

iC_3(_2ft)  =  a;-(-6&)  =  a;  +  66  [P.  4]. 

16.  Since  a;  —  3  (— 2^)  =a;  + 6  J  [Ex.  15],  what  is  the 
value  of  —  3  (—  2  Z>)  ?  Then  a  negative  quantity  multiplied 
by  a  negative  quantity  gives  what  kind  of  quantity  ? 

18.  Since  a  positive  quantity  multiplied  by  a  positive 
quantity  gives  a  positive  quantity  [Ex.  14],  and  a  negative 
quantity  multiplied  by  a  negative  quantity  gives  a  positive 
quantity  [Ex.  16],  we  have, 

Prin.  5, — The  product  of  two  quantities  with  liTce  signs 
is  positive. 

17.  What  is  the  value  of  a;  +  3  (—  2  3)  ? 
Solution :   3  (-  2  &)  =  —  6  & ;  hence, 

x  +  d{-2b)  =  x  +  {-6b)  =  x-6b  [P.  3J. 

18.  Since  x-^3(-2h)  =:x  -  6lf  [Ex.  17],  what  is  the 
value  of  +  3  (—  2  J)  ?  Then  a  negative  quantity  multiplied 
by  a  positive  quantity  gives  what  kind  of  quantity  ? 

19.  What  is  the  value  of  a;  —  3  (+  2  5)  ? 
Solution :   3(+2&)=  +  66;  hence, 

x-S{+2b)z=x-{+Gb)  =  x-6b  [P.  4]. 

20.  Since  x  —  3{-^2b)  =  x  —  6b  [Ex.  19],  what  is  the 
value  of  —  3  (-j-  2  3)  ?  Then  a  positive  quantity  multiplied 
by  a  negative  quantity  gives  what  kind  of  quantity  ? 

19.  Since  a  negative  quantity  multiplied  by  a  positive 
quantity  gives  a  negative  quantity  [Ex.  18],  and  a  positive 
quantity  multiplied  by  a  negative  quantity  gives  a  negative 
quantity  [Ex.  20],  we  have, 

JPrin.  6. — TTie  product  of  two  quantities  with  unlike 
signs  is  negative. 

Note. — Principles  5  and  6  may  be  stated  in  one,  as  follows:  In 
multiplication,  like  signs  give  plus^  and  unlike  signs  minus. 


28  ELEMENTARY  ALGEBRA. 

SIGHT      EXERC  1  SE. 

Name  the  products  of  the  following  quantities,  reciting 
in  each  case  the  proper  principle  of  signs  : 

1-  (+3)x(+4)  7.  (-3)x(+2/) 

2.  (-3)x(+4)  8.  (-5)x(-7) 

3.  (-3)x(-4)  9.  (-V,)x(-V3) 

4.  (+3)x(-4)  10.  (_%)x(+%) 
6.{-i-a)x{-x)  11.  (+73)x(-%) 
6.  (-  a)  X  (+  x)  12.  (-  x')  X  (-  y^) 


Principle  of  Exponents,  etc. 

EXERCISE     12. 

1.  Find  the  product  of  «*  times  aK 
Solution  :a;^  =  axaxaxa 

a^  =  axaxaxaxa 
.'.  a^xa^  =  axaxaxaxaxaxaxaxa  =  aK 

Find  the  product  of 

2.  aP  Xa^         4.  x^  X3^  6.  a^  X  a  8.  r^  X  r^ 

3.  x^  X  x^  5.  TYv'  X  m^  1.  a}  Xa^  9,  y^  X  y^ 

20.  Since  a'^xa^  —  a^  =  «*+^  [Ex.  1],-  we  have, 
Trin,  7. — The  exponent  of  a  factor  in  the  product  equals 
the  sum  of  its  exponents  in  the  multiplicand  and  multi- 
plier, 

4.  Which  is  the  greatest,  axhc,  {aXh)X  c,  oi  h  X 
(ax  c) 

1.  When  a  =  +  3,  J  =  -  2,  and  c  =  -  3  ? 

2.  When  «  =  +  2/3,  !?=-%,  and  <?  =  -  1  ? 

Since  aX  bc  =  {aX  b)  X  c  =  b  x  (aX  c)  for  any  values 
of  «,  5,  and  c  [Ex.  4],  we  have, 

Prin,  8, — Multiplying  one  factor  of  a  quantity  multi- 
plies the  quantity. 


ALGEBRAIC  MULTIPLICATION,  29 

SIGHT      EXERCISE. 

Name  the  products  in  the  following  examples  in  ac- 
cordance with  the  principles  of  multiplication  : 

1.  axxxy  12,  a^  X  aJ^y 

2.  mxaxb  13.  (—  x-)  X  (+  ma^) 

3.  zxxxn  14.  (+  r)  X{—ny^) 

4.  mxcxa  15.  (—  x^)  X  {—x^y^) 
b,  aXhy  16.  dxX2y 

6.  (+S)X{+2y)  17.  (2x)x(-Sy) 

7.  (+5)x(-3ir)  18.  (-2x)x{+3y) 

8.  (-4)x(-2c)  19.  (-2x)x{-3y) 

9.  (—  a)  X  {—be)  20.  xy  X  xy 

10.  (-«)  X  (+xy)  21.  i-xy)  X  (a^) 

11.  aXax  22.  (—  a;2y)  X  (—  xy^) 

Definitions. 

21.  The  process  of  taking  one  algebraic  quantity  as 
many  times,  and  in  such  a  manner,  as  is  indicated  by  an- 
other, is  Algebraic  Multiplication, 

22.  The  quantity  taken  is  the  Multiplicand. 

23.  The  quantity  which  shows  how  many  times  and  in 

what  manner  the  multiplicand  is  taken  is  the  Multiplier, 

Bemark.— The  sign  of  the  multiplier  shows  in  what  manner  the 
multiplicand  is  taken,  whether  additively  or  subtractively. 

24.  The  result  obtained  by  algebraic  multiplication  is 
the  Algebraic  Product. 

25.  The  Arithmetical  Product  of  two  quantities  is 
their  product  irrespective  of  sign.  It  is  the  result  obtained 
by  taking  one  arithmetical  quantity  as  many  times  as  there 
are  units  in  another. 


30  ELEMENTARY  ALGEBRA. 

Problem  1.    To  multiply  a  monomial  by  a  monomial. 

Illustration.— ]\Iultiply  -Yba^¥(^d  hj  —^a^h^c. 

Solution :  Since  multiplying  one  fac- 
tor of  a  quantity  multiplies  the  quantity  Form. 
[P.  8],  +  5  X  a*  X  J*  X  c^  X  tZ  is  multi-            -\-ha^h'^(^  d 

plied  by  —  3  X  a^  X  &*  X  c,  if  +  5  is  mul-  q    zjfi 

tiplied  by  —  3,  a^  by  a^,  &*  by  h^,  c^  by  c, 


and  dhjl.    -  3  times  +  5  is  -  15  [P.  6] ;  —  1^  oJ^V^  (?d 

a?  times  a^  is  a^,  h^  times  i*  is  &',  and 

c  times  c^  is  c^  [P.  7],  and  1  times  d  \^  d\  hence,  the  product  is 

-l^aJ'h'^c^d. 

From  the  above  explanation  we  derive  the  following 
^ule  1. — Tahe  the  product  of  the  numerical  coefficients, 
annex  to  it  all  the  different  literal  factors  used,  giving  each 
an  exponent  equal  to  the  sum  of  its  exponents  in  the  multi- 
plicand and  multiplier, 

EXERCISE     13. 

Multiply 

1.  +2flj  by  +3flj  9.  —%:^z  by  ^xy^z 

2.  -2x  by  +4a;  10.  7a¥  by  -SaH^ 

3.  —  5y  hj  —2y  il.  —  4 w 7^^  by  —  7 « m^ 

4.  +  6m  by  —3m  12.  dx^y^z  by  —Qxz^ 

5.  +4:x^  by  -^da^  13.  (a  +  by  by  (a  +  if 

6.  —da^  hj  -\-4:a^  14.  (m  —  ^^)*  by  (m  —  nf 
1.  3ahj  2b  15.  3  (a  -  bf  by  4(a  -  bf 
8.  6xg  by  Sxf  16.  a^{a  +  by  by  a^a  +  bf 

Find  the  value  of 

17.  dpqX4: mp^  X  —2rq^     22.  m n^  X  S m^ n^  X  4:m^ n 

18.  -4:r^sX6s^zX7rz^       23.  (7^:2^2)  (3^^)  (4^3^) 

19.  7a^bX-2ab^X-aH^    24.  («  +  J)^ (a  +  Z')^ («  +  ^) 

20.  (2 ab){-Z a b) (-  a b)     25.  (« - 3 Z>)*(a -3bf(a-3  bf 

21.  (-  a:)  (-  a;2)  (-  x')  26.  3  a  (a:  -  y)^  {x  -  yf  {x  -  yf 


ALGEBRAIC  MULTIPLICATION,  31 

Problem  2.    To  multiply  a  polynomial  by  a  monomial. 

EXERCISE     14. 

1.  Which  is  the  greater,  a{b-\-c  —  d)  or  ah-\-ac  —  ad 

1.  When  a  =  3,  5  =  4,  c  =  5,  and  6?  =  6  ? 

2.  When  «  =  +4,  5  =  + 5,  c=-3,  and  ^=  +  5? 

3.  When  «  =  +%,  *  =  +V4,  c=-lV2,  and 

e?  =  +6%? 

26.  Since  «(J-|-c  —  (?)  =  aJ-f-ac  —  «rf  for  any  values 
of  a,  h,  c,  and  e?  [Ex.  1],  we  have, 

Prin,  9. — Multiplying  every  term  of  a  quantity  multi- 
plies  the  quantity, 

SIGHT      EXERCISE. 

Name  the  products  in  the  following  examples  : 

1.  x(a-\-b-\-c)  9.  +  4  (+  3  a;  —  2  ?/  +  3  2) 

2.  y{x-\-y-\-z)  10.  — 2(-3«  +  6&  — 42;) 
Z,  z{x  —  y  —  z)  11.  —5(+2a;  — 3^  +  42;) 

4.  m^a^-¥^(^)  12.  -8(-a;2_2^2_322) 

5.  —  c («  —  5  +  c)  I'i,  ah {x -\- y  —  z) 

6.  +3^{x^-x^-\-l)  14.  2:3^(2;  — y  +  ;2) 

7.  a:3(flfa;2-|_Ja;-|-c)  15.  2  a;  (a;^  +  a:  -  1) 

8.  m^  («  w^  —  5  m  —  w)  16.  —  4:X{x^  ^  x^  —  x) 

WRITTEN      EXERCISES. 

Illustration.— Multiply  2a^ —  Sah  +  6b^  hy  —Sab. 

Solution :  Since   multiplying 
every  term  of  a  quantity  multi-  Form, 

plies  the  quantity  [P.  9],  we  mul-  2fl^  —  3«5  +  6^ 


Sab 


tiply  each   term  of   the  multi- 
plicand  by   —Sab,  and  obtain 
~GaH  +  9aH^-  18ab\  There-        -  Q aH -^  9 aH^  -  ISaP 
fore, 

Rule  2, — Multiply  each  term  of  the  polynomial  by  the 
mo7wmial,  bearing  in  mind  the  principles  of  signs. 


32  ELEMENTARY  ALGEBRA. 


EXERCISE     la 

Multiply 
1.  2a-35  +  4c  by  ha 


2.  aJ'-^ab  +  b^  by  -3^5 

3.  5a;2  +  3a;«/  — 2/  by  bx^y^ 

4.  7a^a:^  — 6aa;y  +  9fl^^^  by  '^axy 

5.  x^  —  xy  —  y^  hj  —2x^y^ 

6.  duH^x  +  5aH^xy-'7ab^y^  by  Sir^.y* 

7.  6  m*  +  5  ??i^  —  4  ?7i^  +  3  m  —  5  by  5  m* 

8-  a5-fl^*^4-«'^'-«^^  +  «^*-^'  by  aH^ 

9.  6:r*^-5a;=^/  +  7aj2«/^-5a;2/'4-5/  by  -6a^xy 

10.  a  (a;  +  2/)  +  ^  (^^  +  ^)  —  ^  (i>  +  $')  by  aic 
IL  («  +  5):r2_(^_^)^,^_|.^j^2  ]^^  2a^y^ 

12.  3«(z  +  2/)-4^»(:i;  +  «/)2-2c(:r  +  ?/)3  by  (iC  +  2/)^ 

13.  2x(a-^h)-3y(a  +  iy-\-4.z{a-\-I?y  by2(a  +  ^f 

14.  ^2  (^  _  ^)3  _  ^  ^,  (^  _  ^)3  +  ^2  (^  _  ^)4  by  a  J  (a;  -  yf 

15.  px(p  +  q)-qx{p  +  qf  +i?  §'  (i?  +  ^)'  by 

pqx(p  +  qY 


Algebraic  Division. 

Principles  of  Signs. 

EXERCISE     16- 

1.  By  what  algebraic  number  must  -f-  4  be  multiplied  to 
produce  +  1^  ?  Theu,  +  1^  divided  by  +  4  will  give  what 
algebraic  number  ?  Then,  the  quotient  of  two  positive 
quantities  is  what  kind  of  quantity  ? 

2.  By  what  algebraic  number  must  —  4  be  multiplied  to 
produce  —  12  ?  Then,  —  12  divided  by  —  4  equals  what 
algebraic  number  ?  Then,  the  quotient  of  two  negative 
quantities  is  what  kind  of  quantity  ? 


ALGEBRAIC  DIVISION.  33 

27.  Since  the  quotient  of  two  positive  quantities  is  a 
positive  quantity  [Ex.  1],  and  the  quotient  of  two  nega- 
tive quantities  is  a  positive  quantity  [Ex.  3],  we  have, 

Prin,  10. — The  quotient  of  two  quantities  with  like 
signs  is  positive. 

3.  By  what  algebraic  number  must  +  4  be  multiplied  to 
produce  —  12  ?  Then,  —  12  divided  by  -[-  4  equals  what 
algebraic  number?  Then,  the  quotient  of  a  negative  quan- 
tity divided  by  a  positive  quantity  is  what  kind  of  quantity  ? 

4.  By  what  algebraic  number  must  +  4  be  multiplied  to 
produce  -|- 1^  ?  Then,  -f-  12  divided  by  —  4  equals  what 
algebraic  number  ?  Then,  the  quotient  of  a  positive  quan- 
tity divided  by  a  negative  quantity  is  what  kind  of  quantity  ? 

28.  Since  a  negative  quantity  divided  by  a  positive 
quantity  gives  a  negative  quantity  [Ex.  3],  and  a  positive 
quantity  divided  by  a  negative  quantity  gives  a  negative 
quantity  [Ex.  4],  we  have, 

Prin,  11, — The  quotient  of  two  quantities  with  unlike 
signs  is  negative. 

Note. — Principles  10  ancl  11  may  be  stated  in  one,  as  follows :  In 
division,  like  signs  give  plus  and  vmlike  signs  minvts. 

SIGHT     EXERCISE. 

Name  the  quotients  in  the  following  examples  : 

1.  (+13)--(+3)  9.  (+73)H-(-%) 

2.  ( -  18)  ^  (+  6)  10.  (-  3  %)  -  (+  'A) 

3.  (+  24)  -  (-  4)  11.  (4-  6 'A)  ^  (+  3  Vs) 

4.  (-  36)  -  (-  6)  12.  (-  37 %)  -r  (-  6 'A) 
6.(+xy)^(+x)  13.  (+8  a:) -^(+4) 
^■{-xy)-i-(-y)  14.  (-8  a.)  ^(-2) 

7.  (-xy)  +  (+a;)  X6.  (_9ar=).^(+3) 

8.  (+  xy)  -  (-  y)  16.  (-  6  x>)  -f-  (-  3) 


34:  ELEMENTARY  ALGEBRA, 

Principles  of  Exponents. 

EXERCISE     17. 

1.  By  what  quantity  must  a^  be  multiplied  to  produce 
a^  ?    Then,  a^  divided  by  w*  equals  what  quantity  ? 

2.  By  what  quantity  must  x^  be  multiplied  to  produce 
x^^  ?    Then,  x^^  divided  by  a^  equals  what  quantity  ? 

29.  Since  a^-^a^  =  a^  [Ex.  1],  and  x'^-i-a^  =  a^  [Ex.  2], 
we  have, 

PHn,  12, — The  exponent  of  a  factor  in  the  quotient 
equals  the  difference  of  the  exponents  of  the  factor  in  the 
dividend  and  divisor. 

30.  a^-^a^^  a'  [P.  12]. 

But  a^-^a^  =  1,  since  lXa^  =  a^, 
ftO  =  1.     Therefore, 

JPrin,  13, — Any  quantity  with  an  exponent  of  zero 
equals  unity. 

SIGHT      EXERCI  SE. 

Name  the  quotients  in  the  following  examples  : 

9.   (_j.a;i8)^(-a;i2) 

10.  {- x}') -^  {- x') 

11.  (+3^) --(-32) 

12.  (4-5^)-^(+52) 

EXERCISE     18. 

1.  What  is  the  value  of  54  -^  3  ?  What,  then,  is  the 
value  of  (9  X  6)  -V-  3  ?  Is  (9  X  6)  -r-  3  =  (9  ^  3)  X  (6  -f-  3)  ? 
Is  (9  X  6)  -^  3  =  (9  ~  3)  X  6  ?  Is  (9  X  6)  -^  3  =  9  X 
(6-^3)? 

2.  Which  is  the  greatest,  ai  -r-  c,  {a  -i-  c)  X  b,  or 
ax{b-^c) 

1.  If  a  =  8,  5  =  6,  and  c  =  2  ? 

2.  If  «  =  +  12,  J  =  -  8,  and  c  =  -  4  ? 


1.  a^^  -^  a« 

6.  y^-^y 

2.  a^^  -^  d^ 

6.  z'  -^  z^ 

3.  a^-i-a^ 

7.  (+a;^)^(+^) 

4.  m^'  ~  mi« 

8.  (_a;io)^(^^5) 

ALGEBRAIC  DIVISION,  36 

SIGHT      EXERCISE. 

31.  Since  ah-^c  =  {a-^  c)  xh  =^  ax  {1)-^  c)  for  any 
values  of  a,  J,  and  c  [Ex.  2],  we  have, 

Prin»  14, — Dividing  one  factor  of  a  quantity  divides 
the  quantity, 

SIGHT      EXERCISE. 

Name  the  quotients  in  the  following  examples  : 
i.al)c-i-a  9.  (^-6«)-^(^-3) 

2.  xyz-T-y  10.  (-12:r2)^(_4) 

Z,pqr-^r  ii.  (+15  2)-^(-5) 

4.  a^aj-T-a  12.  (-18  m) -^(+6) 
f^.xf.^y  13.  (-^y)-^(r-^) 
6.  m'n^  -^n-  14.  {-{- a^  z) -^  (-  z) 
n,  a^z-^x  15.  (-:z^y')-^(+«/^) 

5.  as^-r-a^  16.  (+  r  s^)  -r-  (—  s) 

Definitions. 

32.  The  process  of  finding  how  many  times,  and  in 
what  manner,  one  of  two  algebraic  quantities  must  be 
taken  to  produce  the  other  is  Algebraic  Division. 

33.  The  general  problem  of  division  is:  ^^ Given  the 
product  of  two  factors  and  one  of  them,  to  find  the  other, ^^ 

34.  The  quantity  to  be  produced,  and  corresponding  to 
the  product,  is  the  Dividend. 

35.  The  quantity  taken  to  produce  the  dividend,  and 
corresponding  to  the  given  factor,  is  the  Divisor. 

36.  The  quantity  which  shows  how  many  times,  and  in 
what  manner,  the  divisor  must  be  taken  to  produce  the 
dividend,  and  corresponding  to  the  required  factor,  is  the 
Quotient. 


36  ELEMENTARY  ALGEBRA. 

Problem  1.    To  divide  a  monomial  by  a  monomiaL 

Ulustration.— Divide  —Z^a'^hU  by  +8  a^  W. 

Solution  :  Since  dividing  one  fac- 
tor of  a  quantity  divides  the  quantity  Form. 
[P.  14]  -  32  X  a«  X  *»  X  0  is  divided          ^  g  „,  J^^gg^ijs^ 

by  +  8  X  a^  X  h\  if  —  32  is  divided  '■ a    zi3, 

by  +  8,  a^  by  a\  ¥  by  h\  and  c  by  1.  —    ^a  b  c 

-  32  divided  by  +  8  is  -  4  [P.  11] ; 

«^  divided  by  a^  is  a^,  and  h^  divided  by  h^  is  h^  [P.  12] ;  and  c  divided 
by  1  is  c ;  hence,  the  quotient  is  —  4  a^  &^  c.    Therefore, 

Mule  1, — Divide  the  numerical  coefficient  of  the  divi- 
dend by  that  of  the  divisor ;  annex  to  the  quotient  all  the 
different  literal  factors  that  occur  in  the  dividend,  and 
give  to  each  an  exponent  equal  to  its  exponent  in  the  divi- 
dend diminished  hy  its  exponent  in  the  divisor. 

Note, — Two  equal  literal  factors,  one  in  the  dividend  and  one  in 
the  divisor,  may  be  canceled,  since  their  quotient  is  one. 

EXERCISE     19. 

Divide 

1.  6«3by3«2  Z.  -ISa^hHhY  ^ahc 

2.  12«2j  by  -4«  4.  l^a^y^z  by  ^x^y'^z 

5.  —  21  m^  n^  y^  hy  — '^  m"^  n^  y 

6.  maH^x^f  by  -^a^a^y^ 

7.  —mx^f^  by  —12x^y^ 

8.  —^^ax^y^z  by  —  26 aa^yz 

9.  («  +  by  by  {a  +  by 

10.  —  (m  —  ny  by  {m  —  ny 

11.  6a^a-{-xy  by  2a(a-\-xy 

12.  21xy{a-by  by  7x{a-by 

13.  -26z^x-i-yy  by  -6z{x-\~yy 

14.  36  0^  (x^  -  y^y^  by  -  9  rz;^  (^2  _  ^2)7 

15.  -  60  m'  (a^  +  ^>3)8  by  10  m^  (a^  -j-  53)6 


ALGEBRAIC  DIVISION.  87 

Find  the  value  of 

16.  {^3^y^zxQxy^z^)-^8a:^fz 

17.  (-9a:^z^-^d3^z)  X  2xyz^ 

18.  4^3^,2^  X  (-SaH'c*-^2aH^c^) 

19.  (4:a^fz-i-23^y)-{12x^yH^-r-Sa^fz) 

20.  (-4:aH''cXSaH^(^d)  +  {lSaH^(^d-i-6aH^c^) 

21.  (-12a'P(^d-T-4:a:'b(fd)  +  (21a^I)^c^d-T-7a''Pcd) 

Problem  2.    To  divide  a  polynomial  by  a  monomial. 

EXERCISE    20. 

1.  Which  is  the  greater,  (ab-{-bc  — id) -i-b  or  a-{-c  —  d 

1.  When  a  =  S,  5  =  6,  c  =  5,  and  <?  =  2  ? 

2.  When  «  =  +  9,  5=  —  6,  c=+3,  and  d=-\-4t? 

3.  When  a  =  V3,  ^  =  %,  c  =  V6,  and  ^  =  V2? 

37.  Since  (ab  -\-  b c  —  b d)  -r-  b  =  a  -}-  c  —  d  for  any 
values  of  a,  5,  c,  and  d  [Ex.  1],  we  have, 

I'rin,  15, — Dividing  every  term  of  a  quantity  divides 
the  quantity, 

SIGHT     EXERCISE. 

Divide  at  sight : 

1.  (3a;  +  62/-92;)-T-3 

2.  (4a:2-8a;y  +  62/2)-^2 

3.  {10z-\-2(iy-Z0x)-T-10 

4.  (-25a;  +  30y-15;2)H-(-5) 

5.  (16a:2_3^y_j_202^2)^(_|_4) 

6.  (a:^  +  ic^  +  a;)  -T-  a; 

7.  (m^  —  ?/i*  +  ?w^)  -i-  m^ 

9.  (-^'  +  /-3y^H-22^2)^(-/) 
10.  {—  mn^  —  n^r  —  n^ s)  -T-  (—  ?^^) 


38  ELEMENTARY  ALGEBRA. 

WRITTEN      EXERCISE. 

niustration.— Divide  ^aH -^.aH'^ -\-%a¥  h^  2ab, 

Solution :  Since  dividing 
every  term  of  a  quantity  di- 
vides the  quantity  [P.  15],  ^o™' 
we  divide  each  term  of  the          %al)  )  S  a^ b  —  4: a^ b^  -]-  6  aP 
dividend  by  2  a  &  and  ob-                      4:a  —  2  ab -\- 3¥ 
tain  for  the  quotient  4  a  — 
2ab  +  db\    Therefore, 

Mule  2, — Divide  each  term  of  the  polynomial  by  the 
monomial,  bearing  in  mind  the  principles  of  signs. 

EXERCISE    21. 

Divide 

1.  a^-\-a^  by  a  4.  4:X^y^-^%Q?y^  by  2xy 

2.  3^*  +  6a:2  ^  3^  5^  4oaH-Qa¥  by  2ah 

3.  4  a^  —  6  a  5  by  2  a  6.  a^-{- a^  —  a^  —  a  \ij  a 

7.  4a;3  +  6a;2  +  8:r  by  2x 

8.  Qa^-^a^b-\-Za^b  by  Sa^ 

9.  l%ab-l^¥  +  ZOb  by  -Qb 

10.  8«2^*-13a^J3  +  28a*J2  by  4^2 J2 

11.  —^aoi?y-\-Qax^y^-{-^axy^  by  —Zaxy 

12.  a^ x^  —  a^ :ii? -\- a^ a?  —  a X  by  —ax 

13.  14^3 7^*  — 21-^2 7^3  +  28mw2_35^2^  l^y  7^^ 

14.  6a3(^-j-j)-f  9^2(^_|_^)_12«(jt?  +  2')  by  3« 

15.  {a  +  a;)*  -  («  +  a:)^  +  (a  +  a:)^  by  (a  +  ^f 

16.  2«(a  +  ^)*-35(^  +  J)*  +  4c(«  +  ^)*  by  (a  +  5)* 

17.  a;(2:2  +  /)+^(^  +  ^')-^(^'  +  /)  by  ~(.'i:^  +  /) 

18.  x?y^{x  —  yY  —  o^  y  {x  —  yY  -\-x  y^  {x  —  yY  by 

19.  a^  (a  +  J)5  -  «2iz;2  (^  _!_  J)4  ^  ^3^2  (^  _^  J)3   i^y 


SIMPLE  NUMERICAL  EQUATIONS,  39 

Simple  Numerical  Equations. 

EXERCISE    22. 

1.  4  2;  +  3  a*  =  what,  when  ic  =  2  ? 

2.  6  a;  —  2  a;  =  what,  when  ar  =  3  ? 

Expressions  like  4  a;  +  3  a;  =  14   and    6  a;  —  2  a;  =  13   are   called 

3.  Complete  the  following  equations  when  a*  =  3  : 

1.  a;  +  4a;-3a;=  4.  3rz;  +  7  =  8a;- 

2.  3.^-40:4-22:=  5.  3a;+(  )  =  5a;  — 2 

3.  8a;  — 3a;  =  a;+  6.  5a;+6  =  (  )a; 

4.  Complete  the  following  equations  when  a;  =  —  4  : 

1.  7a;-6a;  +  2=:  3.   -  5a:  =  3a;  -  4a:  + 

2.  6a;-8-f  5  =  2a;—         4.  10  a;  -  8a;-|- 7  ==  2a;-f 
The  sign  of  equality  separates  an  equation  into  two  parts,  called 

the  first  and  second  members. 


5.  Complete  the  equation  5a;4-'^  =  3a;+  when  a;  =  4. 
If  we  add  3  to  each  member,  will  the  members  still  be 
equal  ?  If  we  add  x  to  one  member  and  4  to  the  other  ? 
Therefore, 

Prin,  16, — If  the  same  quantity  or  equal  quantities  be 
added  to  equal  quantities,  the  results  will  be  equal. 

6.  Complete  the  equation  3a;  —  5a;4-7  =  4a;—  when 
a;  =  3.  Will  the  equality  of  the  members  be  destroyed  if 
we  subtract  5  from  each  member  ?  If  we  subtract  x  from 
one  member  and  3  from  the  other  ?    Therefore, 

JPrin,  17* — If  the  same  quantity  or  equal  quantities  be 
subtracted  from  equal  quantities^  the  results  will  be  equal, 

7.  Complete  the  equation  5a;  —  3  =  7a;—  when  a;  =  5. 
Will  the  members  remain  equal  if  both  be  multiplied  by  3  ? 
If  one  be  multiplied  by  x  and  the  other  by  5  ?    Therefore, 

JPrin,  18, — If  equal  quantities  be  multiplied  by  the 
same  quantity  or  equal  quantities,  the  results  will  be  equal. 


40  ELE3IENTARY  ALGEBRA. 

8.  Complete  the  equation  dx-{-Qx=l'^x—  when  x  =  3. 
Will  the  members  remain  equal  if  both  be  divided  by  3  ? 
If  one  be  divided  by  x  and  the  other  by  3  ?    Therefore, 

Prin,  19, — If  equal  quantifies  he  divided  hy  the  same 
quantity  or  equal  quantities,  the  results  will  be  equal. 


9.  What  quantity  must  be  added  to  both  members  of 
the  equation  5a;  —  8  =  3a;to  make  it5ir  =  3a;  +  8?  Now, 
what  quantity  must  be  subtracted  from  both  members  of 
hx  =  dx-\-%  to  make  it  5a;  —  3a;=+8?  Are  the  mem- 
bers still  equal  ?    Why  ?    Therefore, 

JPrin,  20, — A  term  may  be  taken  from  one  member  of 
an  equation  to  the  other,  if  its  sign  be  changed. 

Taking  a  term  from  one  member  of  an  equation  to  the  other  is 
called  transposing  it. 

SIGHT      EXERCISE. 

Transpose  the  terms  containing  x  to  the  first  and  the 
others  to  the  second  member  in  the  following  equations  : 

1.  3a:H-5  =  2a;  7.  — 5  +  2a;  =  8-3a; 

2.  3a;  —  5  =  a;-j-7  8.  a;  =  5a;— 7 

3.  9a;  —  8  =  4a;  —  6  9.  a;  +  3  —  5  =  4a; 

4.  8-4a;  =  7-5a;  10.  6a;-7  =  9a;-5 

5.  2a;  +  8  =  7a;-3  11.  a'  =  5a;+7-2a; 

6.  9a;-5  =  8a;  +  2  12.  10a;  =  5-8a;  +  3 

EXERCISE    23. 

X      5      7  a;      2 
1.  If  both  members  of  the  equation  "o  +  "^  =  To"  ~"  3 

be  multiplied  by  12,  a  common  denominator  of  the  frac- 
tions, what  will  the  resulting  equation  be  ?  Will  the 
members  still  be  equal  ?    Why  ?    Therefore, 

Prin,  21, — If  both  members  of  a  fractional  equation  be 
multiplied  by  a  common  denominator  of  its  terms,  it  will 
be  cleared  of  fractions. 


7. 

6x 
12 

7a: 
4 

5 
6 

8. 

3x 

7 

1 
14' 

=  2 

9. 

1 

+1- 

+i' 

1 

~6 

10. 

5 
6^ 

2 
3" 

7 

1 

"18 

SIMPLE  NUMERICAL  EQUATIONS,  41 

SIGHT      EXERCISE. 

Clear  the  following  equations  of  fractions  : 

1.  ^  =  --2 
^23 

X       X   ,   1 

^•4  =  5  +  2 

1,1         1 

3  3 

7         3      5  3         5      7      ,  „ 

^•8^-4  =  2^  "-4^-6  =  8^  +  ^ 

^   3a;      7*      3  ,„    1      ,  o       3  3 

"•-5-- 10  =  5  »^-5*  +  ^  =  i0^--5 

EXERCISE    24. 

1.  If  both  members  of  the  equation  — a;  +  3  =  —  5a;  —  7 
be  multiplied  by  —  1,  what  will  the  resulting  equation  be  ? 
What  change  has  been  made  in  the  terms  ?  Are  the  mem- 
bers still  equal  ?    Why  ?    Therefore, 

JPrin.  22, — If  the  sign  of  every  term  of  an  equation  be 
changed,  the  members  will  still  be  equal. 

Definitions. 

38.  An  Equation,  is  an  expression  of  equality  between 
two  equal  quantities. 

39.  The  Members  of  an  equation  are  the  quantities 
which  are  placed  equal  to  each  other. 

40.  There  are  generally  two  kinds  of  quantities  in  an 
equation,  known  and  unknown. 

41.  A  Known  Quantity  is  one  whose  value  is  given. 

42.  An  Unknown  Quantity  is  one  whose  value  is  to  be 
determined. 


42  ELEMENTARY  ALGEBRA. 

43.  An  Identical  Equation  is  one  that  is  true  for  any 
value  of  the  unknown  quantity ;  as,  ^x  =  x-\-x. 

44.  An  Equation  of  Condition  is  one  that  is  true  only 
for  particular  values  of  the  unknown  quantity;  as,  5  ic  =  30, 
in  which  a;  =  6. 

Axioms. 

45.  An  Axiom  is  a  self-evident  truth. 

46.  The  following  are  the  principal  axioms  of  algebra  : 

1.  Things  which  are  equal  to  the  same  thing  are  equal 
to  each  other. 

2.  If  equals  be  added  to  equals  the  sums  will  be  equal. 

3.  If  equals  be  subtracted  from  equals  the  remainders 
will  be  equal. 

4.  If  equals  be  multiplied  by  equals  the  products  will 
be  equal. 

5.  If  equals  be  divided  by  equals  the  quotients  will  be 
equal. 

6.  Equal  powers  of  equal  quantities  are  equal. 

7.  Equal  roots  of  equal  quantities  are  equal ;  positive 
equal  to  positive,  and  negative  equal  to  negative. 

8.  A  quantity  equally  increased  and  diminished  equals 
the  quantity  itself. 

9.  The  whole  is  equal  to  the  sum  of  all  its  parts. 


Solution  of  Simple  Numerical  Equations. 

lUustration. — 1.  Find  the  value  of  x  in  the  equation 
5a;-8  =  3a;-4. 

Solution  :  Given  5a:  —  8  =  3a:  —  4  (A) 

Transpose  3  a;  to  the  first  member  and  —8  to  the  second  and 
change  their  signs  [P.  20],  then 

5a;-3a;=:-4  +  8  (1) 

Collect  terms,  2x=  4  (3) 

Divide  by  3  [P.  19],  x=  3 


SIMPLE  NUMERICAL  EQUATIONS.  43 

2.  Find  the  value  of  x  in  -rr  —  7  =  77^  —  9. 

o  0 

2  7 

Solution  :  Given  — a:  —  7  =  — a;  —  9  (A) 

3  6 

Multiply  both  members  by  6  to  clear  of  fractions  [P.  21], 
4^ -42  =  7a; -54  (1) 

Transpose  7  a;  to  the  first  member  and  —  42  to  the  second  member 
and  change  their  signs  [P.  20], 

4a;-7a;  =  -54  +  42  (2) 

Collect  terms,  -  3  a;  =  -  12  (3) 

Divide  by  -  3  [P.  19],  x=       4 

Proof :  Put  4  for  x  in  equation  (A), 

2  7 

—  x4—    7  =  — x4  —  9,    or 

3  6 

7  =  — —  9,  or 

3  3 

-4V3  =  -4V3 

47.  Finding  the  value  of  the  unknown  quantity  in  an 
equation  is  called  solving  or  reducing  the  equation, 

48.  Proving  an  answer  obtained  by  solving  an  equation 
is  called  verifying  the  answer. 

EXERCISE    23. 

Solve  the  following  equations  and  verify  the  answers  : 

11.  -x=.-x-\-20 

2.  5x-Sx  =  4:X-6  6         4 

3.  7a;  +  3  =  9a;~9  ^3573 

'  12.   -rX— —X=—X 7 


13.^2:      ^^x^^-^^^x 
7.  io^5x-x='7x-^2        j4   ^_A  =  5^  +  Ji 


4.  ex  —  4:X=10x  —  24: 
6.  -  8x+ 12  =  14 - 

6.  5a?  — 8  =  7a;  +  2a; 

7.  10-{-5x  —  x='7x 

8.  8a;- 8  + 7  =  6a; +  10 

^a;  +  -a;  +  -a;  =  13  —  q-      3-3 

2         3,7,^  75 

-a:--a:  +  -a;  =  19  i6.-a;--a;=3 


4         6         8         4 

5^      10^'~4  +  20 

1_5         ^ 

8 -'6 '^■^12 


111  5         4      2      ,  ^  „, 

9.  ^a;  +  ^a;  +  ^a;  =  13  15.  6^-3  =  3^  +  1  A 

,^23,7,^  75a:,, 

10.  ^x^a;  —  -a;  +  -a;  =  19  16.  — a;— -a;=-  —  11 


4A  ELEMENTARY  ALGEBRA. 

Concrete  Examples  involving  Equations. 

EXERCISE    26. 

1.  A,  B,  and  C  together  own  98  sheep.  B  owns  twice 
as  many  as  A,  and  C  twice  as  many  as  B.  How  many  does 
each  own  ? 

Solution  :  Let  x  =  A's  number ; 

Since  B  owns  twice  as  many  as  A, 

2a;  =  B's  number;  and 
Since  C  owns  twice  as  many  as  B, 

4a;  =  C's  number. 
Since  they  together  own  98, 
4a;  +  3a;  +  a;  =  98. 
Collect  terms,  7  a;  =  98. 

Divide  by  7,  x  =  14,  A's  number ; 

2  a;  =  28,  B's  number ; 

4  a;  =  56,  C's  number. 

2.  A  and  B  together  own  100  acres  of  land,  and  A  owns 
4  times  as  much  as  B.     How  many  acres  has  each  ? 

3.  The  sum  of  three  numbers  is  360.  The  first  is  4 
times,  and  the  second  5  times,  the  third ;  required  the 
numbers. 

4.  A  has  %  as  much  money  as  B,  and  they  together 
have  $50.     How  much  has  each  ? 

5.  My  house  cost  %,  and  my  barn  %,  as  much  as  my 
farm,  and  they  all  cost  15800  ;  required  the  cost  of  each. 

6.  Divide  108  into  three  such  parts  that  the  second 
shall  be  3  times  as  great  as  the  first,  and  the  third  twice 
as  great  as  the  second. 

7.  Divide  760  acres  of  land  into  three  farms,  such  that 
the  first  shall  be  %  as  large  as  the  second,  and  the  second 
%  as  large  as  the  third. 

8.  John  has  5  times  as  much  money  as  William,  and 
the  difference  of  their  amounts  is  $2000.  How  much  has 
each  ? 

Suggestion.— Let  x  =  Wilham's  sum,  then  will  5  a;  =  John's  sum, 
and  5 a; -a;  =  2000. 


CONCRETE  EXAMPLES.  45 

9.  A  number  increased  by  its  one  half,  its  one  third, 
and  its  one  fourth  is  100.     What  is  the  number  ? 

10.  My  horse  cost  6  times  as  much  as  my  harness,  and 
it  cost  $150  more.     What  was  the  cost  of  each  ? 

11.  Jane's  age  is  Va  of  Mary's,  and  the  difference  of 
their  ages  is  8  years.     What  is  the  age  of  each  ? 

12.  The  difference  of  two  numbers  is  49,  and  the  one  is 
8  times  the  other.     What  are  the  numbers  ? 

13.  The  difference  of  two  fractions  is  Yis,  and  the  one 
is  %6  of  the  other.     What  are  the  fractions  ? 

14.  If  a  number  is  diminished  by  the  sum  of  its  Ya  and 
its  Vi  it  will  be  60.     What  is  the  number  ? 

16.  Nine  times  a  certain  number  exceeds  5  times  the 
number  by  90.     What  is  the  number  ? 

16.  Five  sixths  of  my  age  exceeds  three  fourths  of  it  by 
4  years.     What  is  my  age  ? 

.  17.  Two  fields  contain  48  acres  of  land,  and  the  one  is 
twice  as  large  as  the  other,  lacking  15  acres.  How  many 
acres  in  each  field  ? 

Suggestion. — Let  x  =  the  number  of  acres  in  the  smaller  field,  then 
will  2  a;  —  15  equal  the  number  in  the  larger  field,  and  3  ic  —  15  =  48, 
the  number  in  both  fields. 

18.  John  and  William  together  have  1500,  and  John 
has  150  more  than  William.     How  much  has  each  ? 

19.  Twice  John's  age,  plus  15  years,  equals  his  father's 
age,  and  the  sum  of  their  ages  is  75  years.  What  is  the 
age  of  each  ? 

20.  The  difference  of  two  numbers  is  20,  and  their  sum 
is  100.     What  are  the  numbers  ? 

21.  An  agent  bought  three  houses  for  $3500.  The 
second  was  worth  twice  as  much  as  the  first,  plus  1150, 
and  the  third  twice  as  much  as  the  second,  minus  1100. 
What  was  the  value  of  each  ? 


46  ELEMENTARY  ALGEBRA. 

22.  A  man  divided  $4800  among  his  three  sons  :  to  the 
eldest  he  gave  $250  more  than  to  the  second,  and  to  the 
second  1125  less  than  to  the  third.    What  did  each  receive  ? 

23.  Mr.  Jones  sold  his  horse  for  twice  the  cost,  plus 
$40,  and  gained  1190.     What  was  the  cost  ? 

24.  Mr.  Smith  sold  his  horse  for  Y3  of  the  cost,  +  ^10, 
and  lost  $50.     What  was  the  cost  ? 

25.  A  man  spent  Y3  of  his  money,  +  ^^0,  at  one  time ; 
V4  of  it,  +  $30,  at  another  time ;  and  V5  of  it,  +  $40,  at 
another  time,  and  had  nothing  remaining.  How  much 
had  he  at  first  ? 

Suggestion. — Let  x  =  what  he  had  at  first,  then  will  ^3  ^  +  20 
represent  what  he  spent  the  first  time,  ^/4  a;  +  30  what  he  spent  the 
second  time,  and  Vs  a;  +  40  what  he  spent  the  third  time,  and 
^3  a;  +  20  +  V4  ^  +  30  +  Vs  ^  +  40  =  X,  the  entire  sum. 

26.  A  merchant  increased  his  capital  by  Ys  of  itself  and 
$200  more,  and  found  that  he  had  doubled  it.  What  was 
his  capital  ? 

27.  If  to  three  times  a  certain  number  you  add  ^/g  of 
the  number,  and  increase  the  result  by  20,  it  will  be  4  Ye 
times  the  number.     What  is  the  number  ? 

28.  The  difference  between  9  times  a  number  and  20 
equals  the  difference  between  140  and  the  number.  What 
is  the  number  ? 

29.  Twice  a  certain  number  is  as  much  below  100  as  3 
times  the  number  is  above  100.     What  is  the  number  ? 

30.  Three  times  Mary's  age,  increased  by  10  years,  will 
be  her  age  30  years  hence.     What  is  her  age  ? 

31.  Twice  my  age  10  years  ago  will  be  my  age  10  years 
hence.     What  is  my  age  now  ? 

32.  The  sum  of  two  numbers  is  160,  and  3  times  the 
first  equals  5  times  the  second.     What  are  the  numbers  ? 

Suggestion. — Let  x  =  the  first,  then  will  ^j^  x  =  the  second.  Or, 
let  5 a;  =  the  first,  then  will  ox  =  the  second. 


CONCRETE  EXAMPLES.  47 

33.  A  man  sold  %  of  his  land,  and  then  bought  52^4 
acres,  and  then  had  ^4  as  much  as  he  had  at  first.  How 
much  had  he  at  first  ? 

34.  Six  times  John's  age  equals  4  times  William's  age, 
and  the  sum  of  their  ages  is  25  years.  What  are  their 
ages  ? 

35.  Three  times  the  cost  of  my  house  equals  4  times  the 
cost  of  my  barn,  and  the  barn  cost  $1200  less  than  the 
house.     What  was  the  cost  of  each  ? 

36.  Two  thirds  of  one  number  equals  %  of  another,  and 
the  difference  of  the  numbers  is  45.    What  are  the  numbers  ? 

Suggestion. — Since  ^/g  of  one  number  =  '/<  of  another,  then  13 
times  */3  of  the  first  =  12  times  ^j^  of  the  second,  or  8  times  the  first 
=  9  times  the  second,  or  the  first  =  ^/g  of  the  second.  Let  8  x  =  the 
second,  then  will  9  ic  =  the  first. 

37.  A  farmer  raised  4900  bushels  of  corn  and  wheat ; 
Yj  of  the  quantity  of  corn  equals  Ye  of  the  quantity  of 
wheat.     How  much  of  each  kind  did  he  raise  ? 

38.  Five  sixths  of  the  cost  of  a  horse  equals  79  of  the 
selling-price,  and  the  loss  is  $15.     What  was  the  cost  ? 

39.  A  merchant  sold  an  article  for  60  cents,  and  found 

that  Ya  of  his  gain  was  equivalent  to  Y15  of  the  cost.    What 

was  the  cost  ? 
« 

40.  A  man  agreed  to  work  one  year  for  $96  and  a  cow. 

At  the  end  of  9  months  he  left,  receiving  $65  and  the  cow. 
What  was  the  value  of  the  cow  ? 

41.  The  sum  of  two  numbers  is  28.  Their  difference 
is  Ys  of  the  larger  number.     What  are  the  numbers  ? 

42.  John  sold  a  cow  to  William  for  Ys  more  than  it  cost 
him.  William  sold  it  to  Thomas  for  $36,  which  was  V4 
less  than  it  cost  him.     What  did  it  cost  John  ? 

43.  A,  B,  and  C  have  $270.  One  fourth  of  A's  share 
equals  Ys  of  B's,  and  Ys  of  B's  share  equals  Yo  of  C's. 
How  much  has  each  ? 


48  ELEMENTARY  ALGEBRA. 

Addition  of  Polynomials. 

49.  Illustration. — Since  the  whole  equals  the  sum  of  all 
the  parts,  we  arrange  the  terms  so  that  similar  ones  stand 
in  the  same  column,  then  add  the  columns  separately  and 
combine  the  results  by  Kule  2.     Thus, 


1. 

2. 

^a^-Zah-\-hh^ 

4.(a-{-b)-^6(x-y) 

^a^-^-nal-^^l)^ 

7{a-{-b)-6{x-y) 

-9a2_6aZ»  +  6^»3 

-.6{a  +  b)  +  6{x-y) 

laJ'^^al-ZV 

_2(«  +  J)~8(a;-^) 

^a^-\-%al^W 

4.{a  +  b)-2{x--y) 

EXERCISE    27. 

Add  the  following : 

1. 

2. 

3. 

3a  — 4J                 a 

+     i- 

—    c              X—    y-\-    z 

5«  +  95              %a 

-    b-{-dc           2x+6y-l!z 

-6«  +  7Z>              7« 

+  6&- 

-he            9x-5y-\-6z 

4a  — 85          —3a 

+  5&- 

-6c           9x  —  7y-]-5z 

4. 

5. 

2ab-\-dcd  —  4:ad-{-    e 

—  6a^-{-9y^  —  6xy 

6ab-6cd-]-7ad  +  2e 

Sa^-7f-5xy 

6ab  —  6cd—ead  —  3e 

9a^-{-6y^-6xy 

-8ab  +  *7cd-dad-{-6e 

2x^-9y^-\-'7xy 

-4.ab-\-2cd-\-bad-4.e 

6.  Add  hxy^2z\  3xy-5z%  -1xy-\-2z\ 

4:xy-\-6z^,  and  dxy  —  7z^ 

7.  Add  3m^  +  2mn-{-6n^  1  m'' -3mn-^6n\ 

6m^  —  6mn-}-'7n^,  and  —87n^  —  6mn-\-5n^ 

8.  Add  3ax^-5b^y^-\-Sabxy,  6b^y^  -  ^ao^  -  habxy, 

^abxy-hax^-^Wy^,  3Wy^ -habxy -\-9ax^ 


SUBTRACTION  OF  POLYNOMIALS.  49 

Add  :   9.  3  (re  +  ?/)  +  5  (m  +  w),   -  2  (a;  +  i/)  -  5  (m  +  n), 
7  (a:  +  y)  -  8  (m  +  7^,        9  (r/i  +  w)  -  5  (a;  +  y) 

10.  7a(io  +  (7)  +  6^(i?-(Z),  6«(i?  +  (7)-9^»(jo-^), 
5^,(^_^)  +  7«(j,  +  g.),  Za{p^q)-hh{p-q), 
ei(p-q)-Sa{p  +  q) 


Subtraction  of   Polynomials. 

50.  Rule. — Arrange  the  terms  of  the  subtrahend  so  that 
they  stand  under  like  terms  of  the  minuend;  then  change 
the  sign  of  each  term  of  the  subtrahend,  or  conceive  it 
changed,  and  proceed  as  in  addition  of  polynomials, 

Ulustrations. — 

Examples.  Solutions. 

From  3ar-l-7a:y-2/  Zx^-\-    Ixy-^y"^ 

take  biii?  —  ^xy-\-hy^        —^a?-\-   9xy  —  6y^ 


Difference  =  -2a^-j-Wxy  -  7  y"" 

From        3  x 

—  4:y                      dx—    4i/  +  0 

take        7  x 

-{-6y  —  4.z        —7x—    6y-i-4:Z 

Difference  =  —  4:X  —  10y -\-4:Z 

EXERCISE    28. 

1. 

2. 

From  3a:2_|_2^2 

From  9x'-7xy  +  3y^ 

take  4:X^  —  7  y^ 

take  dx'-Sxy  +  Qy^ 

3. 

4. 

From     9  a  +  6  *  - 

-7c            From  lOm^n^ —  7 mn-\-Qn^ 

take  12a-7J  +  9c              take    4:m^n^-{-Smn-\-9n^ 

6.  6. 

From  Sa:^- 72:^2^-    9y^     From  9y*  -  7^^  + 6y  +   5 
take  ^T^'^-bx'y-ny^       take  8y^  +  5y' -  7y +  1^ 


50  ELEMENTARY  ALGEBRA. 

7.  From  9a:2  +  6a;  +  5  take  8a^+7:r-10 

8.  From  25 a^  —  5 2/^  take  ^a?-\-lxy-\-^y^ 

9.  From  10 m^  —  25  n^  take  8m^ -{-7 mn  — dn^ 

10.  From  :2;3 _|_ ^. _|.  9  take  5a^  -  H x^ -{-2x  -  6 

11.  From  8(2;  +  «/)  +  5(:z;  — «/)  take  7(a;4-^)  — 9(a;  — ?/) 

12.  From  the  sum  of  3x-\-  6y  —  4:Z  and  5a;— 7?/  +  52; 

take  9x  —  7 y -\-5z 

X3.  From  9^  —  3a;i/  +  7^^  take  the  sum  of 

3x^  —  5xy-\-l!y^  and  2a.-^  +  ^^y-"5  2^^ 
14.  From  the  sum  of  9  a^  +  7  a^  —  3  o^  +  5 
and  6  «3  -  5  a^  +  7  «  -  3 
take  the  difference  of  9a^-\-5a^  —  '7a-\-6 
and  3a^  +  5a^-6a-i-9 


Symbols  of  Aggregation. 

Definitions. 
51.  The  symbols  of  aggregation  are  the  parenthesis  ( ) ; 
the  braces  {  }  ;  the  brackets  [  ] ;  and  the  vinculum       . 
They  signify  that  the  quantities  inclosed  by  them  shall  be 
considered  together  as  one  quantity. 

62.  In  a  more  general  sense,  the  term  parenthesis  is 
made  to  include  all  symbols  of  aggregation. 

Principles. 

63.  The  expression  a-{-(b  —  c)  denotes  that  the  quan- 
tity Z*  —  c  is  to  be  added  to  a.  If  the  addition  be  per- 
formed the  result  will  be  a-\-I}  —  c.     Therefore, 

JPrin,  23, — If  a  number  of  terms  are  inclosed  hy  a  paren- 
thesis preceded  hy  plus,  the  symbol  and  the  sign  before  it  may 
be  removed  without  altering  the  value  of  the  expression. 


SYMBOLS  OF  AGGREGATION.  51 

54.  The  expression  a  —  {h  —  c)  denotes  that  the  quan- 
tity J  —  c  is  to  be  subtracted  from  a.  If  the  subtraction 
be  performed,  the  result  will  be  a  —  h-\-c.     Therefore, 

Prin,  24, — If  a  number  of  terms  are  inclosed  hy  a  paren- 
thesis preceded  by  minus,  the  symbol  and  the  sign  before  it  mxiy 
be  removed,  if  the  sign  of  every  term  inclosed  be  changed. 

SIGHT     EXERCISE. 

Name  at  sight  the  equivalents  of  the  following  expres- 
sions, without  parentheses  : 
l-a^  +  (3^-^)  5.  2 +  (5 -3)        9.  a -{-a) 

2.  3x-(2y-\-z)      6.  4-(6-4)      I0.3a  —  (2a  —  a) 

3,  a-\-(-b-c)       7.  3-(4-8)      11.  4:b  +  (Gb  -  3b) 
^a  —  {—b  —  c)       8.  6  — (—4)         i2.2y  —  {—y  —  dy) 


65.  a-i-(b-c)  =  a-{-b-c  [F.  23], 
.-.     a-\-b  —  c  =  a-\-(b  —  c).     Hence, 

Prin.  25. — Any  number  of  terms  may  be  inclosed  by  a 
parenthesis  and  preceded  by  plus,  without  changing  the 
value  of  the  expression. 

66.  a-{b-c)  =  a-b  +  c  \V.  24], 
.*.     a  —  b-\-c  =  a  —  {b^c).     Hence, 

Prin,  26, — Any  number  of  terms  m^y  be  inclosed  by  a 
parenthesis  and  preceded  by  minus,  if  the  sign  of  every 
term  inclosed  be  changed. 

SIGHT     EXE  RCISE. 

Inclose  the  last  two  terms  of  the  following  trinomials 
by  parentheses  preceded  by  plus  when  the  middle  term  is 
positive,  and  by  minus  when  it  is  negative  : 

1.  a-\-b  —  c  b.  m-\-n  -{-p  9.  m  —  2n-{-3t 

2.  a  —  b-\-c  6.  m  —  n-{-p  10.  —  x  —  3y  —  3z 
Z.a  —  b  —  c            l.m  —  n—p             li.y  —  2x-\-3z 

4.  x-{-2y  —  z         Q.  3x  —  2y-\-2z       12.  3z  —  2x-\-y 


52  ELEMENTARY  ALGEBRA. 

Problem  1.    To  simplify  a  parenthetical  expression. 

niustrations. — 1.  Simplify  ^ x -\-  {^x -\-  % x  —  ^ x). 
Solution:  3a; +(4a;  +  2a;-3a;)  =  3a;  + 4a;  +  2a;  — 3a;  [P.23]  =  6a;. 

2.  Simplify  5aJ-(3«5-2a^>  +  7a5). 

Solution:  5a&  — (3a&  — 2a6+7a<5>)  =  5a6  — 3a6  +  2a&  — 7a6 
[P.  24]  = -3a J. 

3.  Simplify  2a-\Za  +  2h-\- {4.a  —  dh -{^a- bb)\^ 
Suggestion. — Remove  the  inner  symbols  continuously  until  all  the 

symbols  are  removed.    Thus, 

2a_[3a  +  25  +  |4a- 3& -(2a- 56)}]  = 
2a  — [3a  +  2&  +  i4a  — 3&  — 2a  +  56[]  = 
2a  —  [3a  +  2&  +  4a  —  3&  —  2a  +  5&]  = 
2a-3a-2&-4a  +  3^>  +  2a-5J  =  -3a-4& 

Note. — The  operation  may  often  be  simplified  by  collecting  the 
terms  inclosed  by  a  symbol  at  the  time  of  removing  it. 

EXERCISE    29. 

Simplify 

1.  ^x-\-{^x—^x-^^x)  4.  Zx  —  '^y  —  {^x—'^y) 

2.  4  6f  —  (3  «^  —  7  «  +  6  «)  5.  5  m  —  (6  m  +  2  m  —  m) 
*3.  «  +  2J  +  (6a~3&)  6.2x-\-{^x-(^x-^x)\ 

7.  2a-  {3a  +  (2a-^>)i 

8.  'Hxy  — {— ^xy-\-^xy  —  '^xy) 

9.  'ix^^y—{4.x—{Zx-\-^y)\ 

10.  ^x^{x-\-y)-{2x-4.y) 

11.  ^xy-{^xy-{-(-'^xy-xy)] 

12.  2  +  [2-12  +  (2)-2}+2] 

13.  (6-5)-{6-(5-6)-5}  +  {(5-6)~(5-6)-5( 
1^.  x-y-\x-{y-{z-x)-y\-z'\-\-{x-y-z) 

15.  \_a-\-{x-\-{a-\-x)-\-a}-\-x'\-\-  [a -\- {x -\- a) -\- x\ 

16.  1  _  [_  1  +  { _  1  _  (_  1  _  r+T  -  1)  -  1 }  -  1] 

17.  x  —  [—x  —  {x-{-x)  —x  —  x—  {x-\-  {x  —  x)}'] 


SYMBOLS  OF  AGGREGATION,  53 

Problem  2.    To  inclose  terms  by  symbols  of  aggregation. 

Illustrations. — 

1.  Express  in  binomial  terms  a  —  h  —  c-\-d. 
Solution  :  a-h-c-\-d  =  {a-h)-{c  —  d)  [P.  25,  26]. 

Note. — It  is  customary  to  place  before  the  symbol  the  sign  of  the 
first  terra  to  be  inclosed,  and  if  this  is  negative  the  signs  of  the  terms 
inclosed  must  be  changed. 

2.  Express  in  trinomial  terms  a-\-h  —  c  —  d—  e  +/. 
Solution  :    a  +  h  — c  — d  — e+f={a  +  h  — c)  —  {d-\-e  —  f)    [P. 

25,  26J. 

3.  Express  in  trinomial  terms  having  the  last  two  terms 
of  each  inclosed  by  a  vinculum, 

Solution  :  3a  —  26  +  5c  —  6rf  +  5e  —  4/=(3a  —  26^-5c)  — 
(6(^_5e+4/)=(3a-  26-5c)  -{(Sd-  5e-4f). 

EXERCISE    30. 

Express  in  binomial  terms  : 

1.  2a-{-3b-{-6c  —  2d  3.  7n-]-n  —p  +  q 

2.  a  —  2b-}-c  —  2d  4.  Sz/i  —  2w  —  4^  +  2^ 

5.  a  —  h-{-c  —  d  —  e-\-f 

6.  2a-db-4.c-]-2d-6e-^(jf 

7.  x  —  y-[-2z  —  3v  —  6u-{-4:W 

8.  5p  —  3q-\-6z  —  4:m-\-2n  —  6r 
Express  in  trinomial  terms  : 

9.  Examples  5,  6,  7,  and  8. 

10.  27n  —  3n-{-4:a  —  Gb-\-7c  —  2d  —  4:e-{-g  —  2h 

11.  4:a-2b-dc-4:d-{-5e-\-6f+'7g-2h-\-4:l 

12.  2p-3q-\-4:r  —  2s-\-6t-{-6u-7v-\-2w  —  6t/ 

Express  in  trinomial  terms,  having  the  last  two  terms 
of  each  inclosed  by  parentheses  : 

13.  Examples  10,  11  and  12. 

14.  X  —  y-\-z  —  m-{-n—p-\-q  —  r  —  8 


Form, 

aJ'-^-ah  -\-W 

a^-aJ)  -^h^ 

a^  +  a'b-^aH^ 

-aH-aH^- 

aW 

-\-aH^-^ 

aP  +  b'' 

54  ELEMENTARY  ALGEBRA. 

Multiplication  by  Polynomials. 

niustration.— Multiply  aJ" -\-al)-\-h''  hy  aJ"  -  ah  +  lK 

Solution  :  a^  —  ab  +  b"^  = 
a''  +  (—  a  &)  +  (+&*) ;  therefore 
a^  —  ab  +  b"^  times  a^  +  ab  +  b^ 
equals  the  sum  of 

1.  a^  times  a^  +  ab  +  b^  = 

2.  —ab  times  a^  +  ab  +  b^  — 

3.  +    &2  times  a^  ^ab  +  ¥  = 

which  =       a^  -\-a^W  -\-h^ 

MtUe  3,— Multiply  the  multiplicand  hy  each  term  of  the 
multiplier  and  take  the  algebraic  sum  of  the  products. 

Note. — For  convenience,  arrange  both  multiplicand  and  multiplier 
according  to  the  ascending  or  descending  powers  of  some  letter  as- 
sumed as  a  leading  letter.     Thus,  a  better  order  of 

(a;2  _  5  +  7a;)  (3a:  +  2a;2  -  5)  is  {x^  +  7a;  -  5)  (2a;2  +  3a;  -  5). 

EXERCISE    31. 

Multiply 

1.  a-\-b  by  a-b  9.  Sa^-\-2b  by  4:a^-eb 

2.  a  +  b  by  a  +  b  lo.  7a^-8b^  by  da^  +  Hb^ 

3.  a  —  b  hj  a  —  b  li.  ac  —  bdhj  by  — dx 
^  2a-{-db  hj  2a-db  12.  cc*  -  a:^  +  1  by  o^  +  1 
6.  xy-{-12  hj  xy  —  6  13.  a^ -{-ab -\-b^  hj  a  — b 
6.  x  —  a  hj  x  —  b  14.  a^-\-2x-{-4o  hj  x  —  2 
1.  b  —  X  hy  c  -\-  X  15.  fl^^  —  «*  +  1  by  «*  +  1 
8.  m^-j-n^  by  m2  +  ^2  ^^   gl-Oc^  +  c*  by  9  +  c^ 

17.  J*-4Z>2c24-l6c*  by  ¥  +  4:(^ 

18.  Sm^-}-2mn  hj  2m^  —  Sn^ 

19.  ex^y^-7y^z^  by  6 a^ y^ -+■  7 y'^ z- 

20.  9a^  +  36a-\-1Uhj  3a-12 

21.  cc^  — a^y-\-xy^  — y^  by  x-\-y 

22.  a^  +  aH  +  a^b^  +  aP  +  b'^  by  a-b 


DIVISION  BY  POLYNOMIALS,  65 

Find  the  value  of 

23.  (30:2^32.^^22^2)  (3^__3^y_^2/) 

24.  (5fl2-f  7aJ  +  4J2)(5a2-7aJ  +  4Z>2) 

25.  (9a:2_|_42.^_^2)(9a;2_4^y  +  2/') 

26.  (2a;2_4a.^6)(^_{.2a;  +  3)(2a:*-4a:2  +  18) 

27.  (a*  +  2a2  5  +  4a2^  +  8aJ=*  +  16J^)(«-2J) 

28.  81a;*-54a;2y  +  3Ga:2y2_24a;i/3^16^)(3a;^2f/) 

29.  (2:*-a^2/2_^^)(,^_^^2^_|.2^2)(^_^^_|_^2) 

30.  (2a:5  +  3a.-*^-2a:32^2_j_4^^3_5^2^_^3y5) 

(3a;2^2a;y  +  3r) 


Division  by  Polynomials. 

Ulnstration.— Divide  a^  +  W  by  a-^h 

Solution  :   «  +  &  is  con- 
tained in  a'  +  ^3  as  many  Form, 

times  as  it  can  be  taken  out  ^    i    j  )  ^3  _|_  j3  /  ^2  _  ^  j  _|_  ^ 
of  it.    a  is  contained  in  a*  ^  ,      9\ 

Or  -T-  d    b 

a?  times;  taking  a»  times  IL — _ 

(a  +  6),  or  a3  +  a»  6  out  of  —a^l)-\-l^ 

o»  +  6'   by  subtracting    it,  —  a^l  —  aW^ 

there   remains   —  o'  6  +  &».  ^J2TI|r  J3~ 

a   is   contained    in    —  a'  6  xg   i    zs 

(—  a  6)  times ;  taking  (—  o  h)  

times  (a  +  6),  or  —  a'  6  —  a  h^ 

out  of  —  a^ 6  +  6'  by  subtracting  it,  there  remains  ah  -\-h^.  a  is  con- 
tained in  aV^,  (+  h)  times;  taking  (+  &)  times  (rt  +  5),  or  aW  -k-  J'  out 
of  db^  +  &'  by  subtracting  it,  nothing  remains.  Therefore,  (a  +  h)  is 
contained  in  (a'  +  &*)>  (a*  —  a  6  +  &*)  times. 

Suggestions. — For  convenience,  arrange  the  terms  of  the  dividend, 
divisor,  and  the  several  remainders,  according  to  the  ascending  or  de- 
scending powers  of  some  letter  assumed  as  tlie  leading  letter.  After 
each  subtraction  do  not  bring  down  any  more  terms  than  will  ho.  needed 
for  the  next  operation.  Always  divide  the  first  term  of  the  dividend, 
or  partial  dividend,  by  the  first  term  of  the  divisor  to  obtain  the  next 
term  of  the  quotient. 


56  ELEMENTARY  ALGEBRA. 

EXERCISE    32. 

Divide 

1.  a2+9a  +  18  by  «  +  3  6.  x^  —  y^  by  x--y 

2.  o;^  —  a;  —  20  by  ic  —  5  l.  a? -{-^1  y^  hj  x-\-^y 

3.  a;2- 12a; +  35  by  a; -7  8.  a«  +  J«  by  aJ'-i-h^ 

4.  a;*  +  4ic2-12  by  a;^  -  2  9.  «*-§*  by  a-J 

5.  x^-\-2xy-\-y^  by  a;  +  2/  10.  8a;3  +  27y^  by  2rr  +  3y 

11.  4a:2_4^__24  by  2a;  +  4 

12.  4  a;^  —  4  a  a;  -—  3  6f^  by  2  ic  +  a 

13.  8rc2-14aa?~15a2  by  4a;  +  3a 

14.  64 a;«  -  125  /  by  4a;3  -  5  / 

15.  d2x^-{-y^  by  2a;  +  y 

16.  8m^-277i«  by  2^3-3^2 

17.  x^ -\- x^ y^ -}- y^  by  a:^  +  a;^  +  ^^ 

18.  x^-{-4:a^-\-lQ  hj  a^-2x-i-4t 

19.  a8  +  a*J*  +  J«  by  a*  +  a2j2_^J* 

20.  16  m*  ^»  +  36  m^  n^  +  81  m^  n!" 

by  4  m^  7^*  —  6  m^  n^  +9  m*  w 

21.  a;*  +  4a;3  +  6rr2-}-5iC  +  2  by  .T2  +  3a;  +  2 

22.  6a;*  +  19a;3  +  10a;2^2a:  +  5  by  2a;  +  5 

23.  5x*  +  2a;3-20a;2-23a;-6  by  5a;2+7a:  +  2 

24.  8a«-16««-34a*  +  32a2-6  by  2a*-7a^2  +  3 

25.  2  a;^  —  3  a;  2f^  +  2/2  +  rr  ;2  —  2/  2;  by  a;  —  ^ 

26.  x^  —  ^xy-\-4.y'^  —  ^z^  by  a;  — 2?/  +  3;2 

27.  4a:2  — 9^/2— 6?/2  — ;22  by  %x-{-Zy-{-z 

28.  4a;2  +  12a;?/  +  9/  — 2;2  by  %x-\-Zy  —  z 

29.  :x^  —  x^y^  —  %xy^  —  'f  by  a:^  +  a; ^  +  2/^ 

30.  fl2  +  2«^  +  Z'2-c2-2cJ-^2  by  «  +  ^>  +  c  +  ^ 


,4^2 


NUMERICAL  EQUATIONS,  67 

Simultaneous   Numerical    Equations  of  Two   Un- 
known Quantities. 

EXERCISE    33. 

1.  What  is  the  value  of  x  in  the  equation  2  a;  +  3  ?/  =  24, 
if  y  =  4  ? 

Solution :  Put  4  for  y  in  the  equation, 

2a; +  12  =  24  (1) 

Transpose  12,  2  a;  =  12  (2) 

Divide  by  2,  a;  =    6 

2.  What  is  the  yalue  of  x  in  the  equation  4  a;  —  3  y  =  26 

1.  If  y  =  1  ?  3.  If  ?/  =  3  ?  5.  If  1/  =  5  ? 

2.  If  2/  =  2  ?  4.  If  y  =  4  ?  6.  If  2/  =  6  ? 

3.  What  is  the  value  of  x  in  7  ic  +  5  ?/  =  66 

1.  If  2^  =  3  ?  3.  If  3^  =  5  ?  5.  If  ^  =  7  ? 

2.  If  2/  =  4  ?  4.  If  1/  =  6  ?  6.  If  3/  =  8  ? 

57.  A  single  equation  containing  two  unknown  quanti- 
ties can  be  satisfied  with  any  number  of  pairs  of  values  of 
the  unknown  quantities. 

4.  What  values  of  x  and  y  will  satisfy- 

both    4:X-Sy  =  26  (A) 

and      1lx  +  6y  =  66?  (B) 

Solution :  Multiply  (A)  by  5  and  (B)  by  3  to  make  the  coefficients 
of  y  numerically  equal  [P.  18], 

20  a:- 15  y  =  130  (1) 

21a;  +  15y  =  198  (2) 

Add  (1)  and  (2)  [P.  16],  41  a;  =  328  (3) 

Divide  by  41,  a;  =     8 

Put  8  for  X  in  (B), 

56  +  5y=   G6 
Transpose  56,  5y=    10 

Divide  by  5,  y  z=     2 

Verify  by  putting  8  for  x  and  2  for  y  in  (A)  and  (B), 
32  -    6  =    26,    which  is  true. 
56  +  10  =   66,    which  is  true, 
a:  =  8  and  y  =  2  are  the  only  values  of  the  unknown  quantities 
that  will  satisfy  both  equations. 


58  ELEMENTARY  ALGEBRA, 

5.  If      2a;  +  3?/  =  18 
and  3  a; +  2?/ =  17 

6  a;  +  9  «/  =  54    Why  ? 

5?/ =  20 

2ir+12  =  18 
2a;=    6 
i?;=    3        '' 

Verify:  Are  3.+  3,  =  18).^^^ 

and  3a;  +  2^  =  17j  ^ 

58.  Two  equations  of  two  unknown  quantities  can  be 
satisfied  only  by  particular  values  of  those  quantities. 

Definitions. 

59.  When  two  equations  express  such  relations  between 
two  or  more  unknown  quantities  that  neither  of  them  can 
be  reduced  to  the  form  of  the  other,  they  are  called  Inde- 
pendent Equations. 

niustration.— 1.  3x-\-y=:l2  and  2x  —  dy  =  6  are  in- 
dependent equations. 

2.  x-\-y  =  6  and  5  tr  -|-  ^  ^  =  ^^  are  not  independent 
of  each  other,  since  the  first  multiplied  by  5  will  give  the 
second. 

60.  If  two  or  more  independent  equations  are  to  be 
satisfied  by  the  same  values  of  the  unknown  quantities, 
they  are  called  Simultaneous  Equations. 

61.  To  solve  two  simultaneous  equations  of  two  un- 
known quantities,  we  first  deduce  from  them  a  single  equa- 
tion containing  only  one  of  the  unknown  quantities.  That 
is,  we  perform  such  operations  upon  the  given  equations  as 
are  necessary  to  get  rid  of  one  of  the  unknown  quantities. 
This  process  is  called  Elimination.  • 


ELIMINATION.  59 

Elimination  by  Addition  or  Subtraction. 

niustration.— Solve    2x-\-3y=l%  (A) 

and     3  a; +  5?/ =  19  (B) 

Solution :  Multiply  (A)  by  3  and  (B)  by  2  to  make  the  coefficients 
of  X  alike  [P.  18], 

6a:  +   92/  =  36  (1) 

Qx  +  \Qy  =  m  (2) 

Subtract  (1)  from  (2)  [P.  17], 

y=  2 
Put  2  for  y  in  (A), 

2ic  +  6  =  12  (3) 

Transpose  6,  2x=   Q  (4) 

Divide  by  2,  x=   S 

Verify  by  putting  3  for  x  and  2  for  y  in  (A)  and  (B), 

m  —  iq  I  ^^^^  ^^  which  are  true. 

Kote. — If  the  signs  of  the  like  terms  in  (1)  and  (2)  were  unlike,  the 
equations  would  have  to  be  added  to  eliminate  x. 


EXERCISE    84. 

Solve : 

1.  3a;  +  4?/  =  29) 
2x-\-Sy=^6  \ 

8. 

3x-4:y=    3) 
5x-3y  =  lQ  ) 

2,  5x  —  3y  =  22\ 

2x-{-9y  =  19  ) 

9. 

5a;+7y  =  0) 
8x  +  6y  =  0\ 

3.  4a;-    7y=-l  ) 
3a;+ll2^  =  48     j 

10. 

6x-\-    92^  =  45) 
Sx-i-16y='70\ 

4.7a:  +  8y  =  2        ) 
6x-2y=-U  ) 

11. 

Ux-{-19y  =  25        ) 
21a;~17?/=-190  j 

6.  3x-{-7y  =  S3) 

12. 

16x-{-l()y  =  105 
10a;-15i/=-G5  ■ 

6.  5a;-7i/  =  13     ) 
'7x-6y=-l\ 

*13. 

^  +  ^  =  36- 

7.     x-\-3y  =  10 
dx  —  5y  =  30 

|.  +  |,  =  25 

Clear  of  fractions  first. 


60  ELEMENTARY  ALGEBRA. 

Concrete  Examples  involving  Simultaneous 
Equations. 

EXERCISE     38. 

1.  A  and  B  together  have  $500 ;  if  A  had  three  times 
and  B  four  times  as  much  as  now,  A  would  have  1800 
more  than  B.     How  much  has  each  ? 

Suggestion.—        Let  x  =  the  number  of  dollars  A  has, 
and  y  =  the  number  of  dollars  B  has. 
Now,  since  they  together  have  $500, 

a;  +  2/  =  500  (A) 

Since  3  times  A's  sum  exceeds  4  times  B's  by  $800, 

3a; -42/ =  800  (B) 

Solve  (A)  and  (B)  to  obtain  results. 

2.  Three  times  A's  age  added  to  twice  B's  equals  85 
years,  and  twice  A's  added  to  three  times  B's  equals  90 
years.     What  is  the  age  of  each  ? 

3.  If  3  apples  and  4  peaches  are  together  worth  10 
cents,  and  5  apples  and  2  peaches  12  cents,  what  are  they 
worth  apiece  ? 

4.  If  4  bushels  of  corn  and  5  bushels  of  oats  together 
weigh  374  pounds,  and  3  bushels  of  corn  weigh  48  pounds 
more  than  4  bushels  of  oats,  what  are  their  respective 
weights  per  bushel  ? 

5.  A  house  and  barn  together  cost  $3000,  and  three 
times  the  cost  of  the  house  exceeds  five  times  the  cost  of 
the  barn  by  IIOOO.     What  is  the  cost  of  each  ? 

6.  If  A's  money  were  increased  by  $36,  he  would  have 
three  times  as  much  as  B  ;  and  if  B's  money  were  dimin- 
ished by  $5,  A  would  have  twice  as  much  as  B.  Find  the 
amount  each  has. 

7.  The  sum  of  two  numbers  is  38,  and  twice  the  less  is 
18  times  their  difference.     What  are  the  numbers  ? 

8.  Five  coins  of  one  kind  and  six  of  another  are  worth 
$4.25,  but  four  of  the  first  kind  and  seven  of  the  second 
are  worth  $4.50.     Required  the  value  of  each  coin. 


CONCRETE  EXAMPLES.  61 

9.  If  7  bushels  of  corn  and  10  bushels  of  oats  are  worth 
$8.20,  and  6  bushels  of  corn  and  8  bushels  of  oats  $6.80, 
what  is  the  price  of  each  per  bushel  ? 

10.  If  8  men  and  12  boys  earn  $168  a  week,  and  9  men 
and  7  boys  $150  in  the  same  time,  what  are  the  daily  wages 
of  each  man  and  boy  ? 

11.  A  drover  sold  12  sheep  and  8  cows  for  $392  ;  had 
he  sold  3  more  cows  and  5  less  sheep,  he  would  have  re- 
ceived $482.     What  was  the  price  of  each  sheep  and  cow  ? 

12.  A  merchant  bought  40  grammars  and  50  readers  for 
$77 ;  had  he  bought  50  grammars  and  40  readers,  they 
would  have  cost  $1  less.    What  was  the  price  of  each  book  ? 

13.  If  8  men  and  6  boys  earn  as  much  per  day  as  9 
men  and  4  boys,  and  the  difference  between  the  daily 
wages  of  a  man  and  a  boy  is  $1,  how  much  does  each  re- 
ceive per  day  ? 

14.  A  grocer  has  two  kinds  of  coffee  :  if  he  mixes  12 
pounds  of  the  first  kind  with  1*8  pounds  of  the  second,  the 
mixture  will  be  worth  20  cents  a  pound ;  but  if  he  mixes 
24  pounds  of  the  first  kind  with  6  pounds  of  the  second, 
the  mixture  will  be  worth  16  cents  a  pound.  What  is  the 
value  per  pound  of  each  grade  ? 

15.  If  A  buys  40  acres  of  land  from  B,  B  will  have 
twice  as  much  as  A ;  but  if  he  buys  80  acres,  they  will 
have  the  same  amount.     How  much  has  each  ? 

16.  C  has  60  acres  of  land :  if  A  buys  C's  land,  A  will 
have  as  much  as  B ;  but  if  B  buys  it,  B  will  have  three 
times  as  much  as  A.     How  many  acres  has  each  ? 

17.  The  sum  of  two  numbers  exceeds  twice  their  differ- 
ence by  30,  and  twice  the  first  equals  three  times  the  sec- 
ond ;  required  the  numbers. 

18.  If  B  were  to  give  A  $25,  they  would  have  equal 
sums  of  money ;  if  A  were  to  give  B  $22,  B  would  have 
twice  as  much  as  A.     How  much  has  each  ? 

4 


62  ELEMENTARY  ALGEBRA. 

Partial  Treatment  of  Algebraic  Involution. 

Definitions. 

62.  The  result  obtained  by  using  a  quantity  two  or 
more  times  as  a  factor  is  a  Power  of  the  quantity. 

63.  The  number  of  times  a  quantity  is  used  as  a  factor 
to  produce  a  power  is  the  Degree  of  the  power. 

Illustration. — Thus,  a^  is  a  power  of  the  fourth  degree 
when  derived  from  a^,  since  a/^  X  a^  X  a^  X  a^  =  a\ 

64.  The  degree  of  a  power  is  expressed  by  a  quantity 
called  an  Exponent,  written  on  the  right  hand  above  the 
quantity. 

Illustration. — The  4th  power  of  a^  is  written  {a^)\  4  is 
the  exponent,  and  denotes  the  degree  of  the  power. 

65.  The  process  of  raising  an  algebraic  quantity  to  any 
power  is  Algebraic  Involution, 

Principles. 

66.  {J^ay=(+a)x{-ha)  =  -ha' 
(-aY  =  (-a)x{-a)  =  -\-a^ 

(+«)*  =  (+«)  X  i+a)  X  (+«)  X  (+«)  =  +a* 
(-  ay  ={-a)x  (-  a)  X  (- a)  X  {- a)  = -{-  a'^ 

In  the  same  way  it  may  be  shown  that  *  ( ±  «)^  =  -j-  a^ ; 
{±aY  =  -{-a^;  (±  «)i«  =  +  «^« ;  etc.     Therefore, 

JPrin.  27 » — An  even  power  of  a  positive  or  a  negative 
quantity  is  positive, 

67.  (+«)3=(+«)x(+a)x(+«)  =  +a^ 
(-  ay  =  {—  a)x{—  a)x(—  a)  =  —  a^ 
{+ciy={-i-a)x{+a).x{+a)x{+a)x{+a)  =  -\-a' 
(—  ay  =  {—  a)X{—  a)x{—  a)x{—  a)X(—  a)  =  —  a^ 

*  ±  a  is  read  plus  or  minus  a. 


ALGEBRAIC  INVOLUTION,  63 

In  a  similar  manner  it  may  be  shown  that 

(±  «)'  =  ±  rt^ ;  (±  a)'  =  ±  a* ;  etc.     Therefore, 
Brin.  28. — An  odd  power  of  a  quantity  has  the  same 
sign  as  the  quantity. 

SIGHT     EXERCISE. 

Give  the  true  values  of  the  following  expressions  : 

1.  (+2)«  6.  (+3f  11.  {±ay 

2.  (-2)2  7.  (-3)3  12.  {-xY'^ 

3.  (+2)»  8.  (-a:)*  13.  (+5)3 

4.  (-2)3  9.{^xf  14.  (-5)3 
6.  (±2)*                10.  (+rt)»  15.  (±5)* 


16. 


(4)'     ■'■(4)'     '-(^J 


68.  {a^Y  =  a^xa^Xa^Xa^  =  a}''  =  a^''\     Therefore, 
Brin,  29, — Multiplying  the  exponent  of  a  factor  by  the 
exponent  of  a  power  raises  the  factor  to  that  power. 

SIGHT    EXERCISE. 

Give  the  true  values  of  the  following  expressions  : 


1.  {a^r 

5.  (-a')' 

9.  (±a')« 

3.(x3r 

6.  (-a*)« 

10.  (-/)» 

3.  (a*)' 

7.  {^^Y 

11.  (-  2=f 

4.  (+aT 

8.  (±a»)* 

12.  (-  Vf 

13.      + 


(DT     -{-i 


69.  (a^l^cY^a^y'c  X  a^h^c  X  a^^'^c  X  a'l^c^ 
a^  X  fl2  X  «2  X  a'  X  ^3  X  J3  X  J3  X  ^,3  X  c  X  c  X  c  X  c  [P.  8]  = 
(a')*  X  {h^Y  X  (c)^     Therefore, 

Prin.  30, — Raising  every  factor  of  a  quantity  to  a 
given  power  raises  the  quantity  to  that  power. 


64  ELEMENTARY  ALGEBRA, 

SI  GHT      EXERC  ISE. 

Give  the  true  values  of  the  following  expressions  : 

1.  (2  a^  hf  5.  (+  a^  h^  c'^f  9.  ( ±  2  a^  h^f 

2.  (2«3J2)3  Q   (_  ^^2^3)4         10^  [-.a}^h^c' 

3.  (3  a*  h^f  7.  ( ±  m^  ^2^5)6        1 1.  (+  m^^  n^  r^ 

4.  (-  2  a^ a;5)2       8.  (+  0.-* 2/^  ;25)5         12.  ( ±  3  w^  0,-2)3 


klO 


13. 


14. 


i^^y^y  15.  (|r*2/^)« 

i^o^f^  16.  (±|«^J^)* 


Problem  1.    To  raise  a  monomial  to  any  power. 

Illustration. — Eaise  —^a^¥  c*  to  the  third  power. 

Solution :   Since  raising 
every  factor  of  a  quantity  Tora^, 

to  a  power  raises  the  quan-  (—  3  «^  Z>2  d^Y  =  —  27  «^  ¥  c^^ 

tity  to  that  power  [P.  30], 

(-'  3  a^  y^  &f  =  (-  3)3  X  (a3)3  x  (62)3  x  (c4)3,  (_  3)3  =  _  27  [P.  28]  ; 
(a3)3_a9,  {b^f  =  h\  and  (c4)3  =  ci2  j-p,  29];  hence,  the  result  is 
-27a»68c«     Therefore, 

Mule, — Eaise  the  numerical  coefficient  to  the  required 
power,  and  multiply  the  exponent  of  each  literal  factor  by 
the  exponent  of  the  power, 

EXERCISE    30. 

Find  the  value  of 

1.  (aHc^y  7.  (2aH^cdy 

2.  {2abHY  8.  {-Sa^y^zy 

3.  {-dan^(^f  9.  {2x'y^z^y 

4.  {-'2xyzY  10.  {omn^z^Y 

5.  {^^x^y^zf  11.  {(a-\-h)  {c^d)]^ 


6.  (— 4m2^^^a;)3  12.  [m ay^ {a -\- h) 


3)4 


ALGEBRAIC  INVOLUTION.  65 

13.  J3(«  +  J)2  (x-yf]''         15.  {2aH^{x--\-ifY\^ 

14.  {a3^c(m  +  w)2}*  16.  \rrv'{x-ijf{x-{-yf\'' 


17. 


(Sa^Z^^c)^  X  {-%ahc'Y  X  (|«'^'^)' 


18.  (9a:2^;2-3a;»2^^2)^-v-(6r^/2)2 

19.  {{^7?yH''y-{^x'f^f\X^x^y^z^ 

20.  j(5a:*y«;z^«)2+(a;2^^6)4|_^2a:*y^2^5 

21.  {a^  W  (^Y""  +  2  («*  J*  c*)«  -  3  (a»  ^>8  c«)3 

22.  (4  7?  y''  zf  X{^xy^  z^f  -  8  {x^  y^  zf 

23.  (2  a;* ^ ;z^)*  ^  (- ^ y^ z^f  X{-2xy''zY 

24.  {(3a:3  2^i2«)«X  (- aa^/;?^}^ -^  (- 3^^'^)^ 

Problem  2.    To  square  a  binomial. 

Principles. 

70.  The  square  of  the  sum  of  a  and  b,  or  (a  +  J)"  = 
{a-^  b)  (a  +  b)  =  a^  +  2ab  +  b^     Therefore, 

JPrin,  31, — The  square  of  the  sum  of  two  quantities 
equals  the  square  of  the  first,  plus  twice  their  product,  plus 
the  square  of  the  second. 

71.  The  square  of  the  difference  of  a  and  b,  or  (a  —  by 

=  (a  -  b)  (a  -  b)  =  a^  -  2ab  +  b^     Therefore, 

Prin.  32. — The  square  of  the  difference  of  two  quan- 
tities equals  the  square  of  the  first,  minus  twice  their  prod- 
uct, plus  the  square  of  the  second. 

Note. — These  two  principles  may  be  stated  together  as  follows: 
''The  square  of  a  bmamial  equals  the  sum  of  the  squares  of  its  terms^ 
and  twice  their  algebraic  product.''^ 

ninstrationg.— Square  2a +  35  and  da^  —  by^. 
Solutions :  (2  a  +  3  &)«  =  (2  a)^  +  2  x  2  a  x  3  6  +  (3  &)«  [P.  31]  = 

4a2  +  12a6  +  96». 
(3a;»  -  52/T  =  (3  a;')'  -2x3a;«x5y2  +  {'iy^f  [P.  32]  = 

9  a;* -30  a;*  2^*  + 25  y. 


66  ELEMENTARY  ALGEBRA. 

EXERCISE    37. 

Find  the  value  of 

i.{x^yy  11.  (%a-\-dbf  21.  {ah-cdf 

2.  (x  —  yf  12.  {ha  —  ^hf  22.  {xy-\-yzf 

3.  {m  +  nf          13.  (2 a;  +  8)^  23.  {2pq-3 rf 
^  (m-  nf           14.  (3x-  5)2  24.  (a^  J  +  a  b^f 

6.  (iP  +  4)2  15.  (5  +  2a:)2  25.  (2  a:^  ^  __  3  ^  ^2^ 

6.  (x  -  7)2  16.  (6  -  3  ^)2  26.  (^+T -  1)2 

7.  (2:  +  «)2  17.  (xy-\-lf  27.  (a  -  6  +  1)^ 

8.  {x  —  af            18.  (a;  ^  +  5)2  28.  {x-\-y-\-  zf 
9.{^x-\-yf          19.  (1-C6?)2  29.  {x-\-y-  zf 


10.  (3  a;  -  2()2         20.  (1  +  a;  ^)2  30.  (a  -  J  +  cf 

Problem  3.    To  cube  a  binomiaL 
Principles. 

72.  The  cube  of  the  sum  of  a  and  i,  or  («  +  ^)^  = 
{a  +  l)){a  +  h){a-^J))  =  a3  +  3«2  j  4. 3a^,2_^  53^  There- 
fore, 

JPrin,  33, — The  cube  of  the  sum  of  two  quantities  equals 
the  cube  of  the  first,  plus  three  times  the  square  of  the  first 
into  the  second,  plus  three  times  the  first  into  the  square  of 
the  second,  plus  the  cube  of  the  second. 

73.  The  cube  of  the  difference  of  a  and  b,  or  {a  —  bf  = 
(a-b){a-b){a-b)  =  a^-3aH^3ab^-  b\  There- 
fore, 

JPrin.  34. — The  cube  of  the  difference  of  two  quantities 
equals  the  cube  of  the  first,  minus  three  times  the  square  of 
the  first  into  the  second,  plus  three  times  the  first  into  the 
square  of  the  second,  minus  the  cube  of  the  second. 

Illustration. — Cube  2x-{-dy. 

Solutions :  {2x  +  Syf  =  {2xf  +  S  X  {2xf  x  (3y)  +  3  x  (2x)  x 

(3y)2  +  (3y)3  [P.  33]  =  8x^  +  dQx^y  +  54:Xy^  +  27y\ 


COMPOSITION,  (57 

EXERCISE    38. 

Find  the  value  of 

1.  (x  -f-  ijf  8.  (2  +  zf  15.  {^^-^  If 

2.  {x  -  Iff  9.  (2:  +  2  zf  16.  (a:^  -  2  y3)3 

3.  (y  +  2)3  10.  (a;  -  2  yf  17.  (2  a;^  -  3  /j^ 
4.(?^-2)3             \i.{ax^l)yf       ■    \^.{x'-?>xyf 

5.  (l+a:f  12.  (2a;  +  a;y)3  19.  (a;^  y  +  a:  ?/2)3 

6.  {x  -  \f  13.  (a-  -  x^f  20.  (3  a:2  -  4  2/^)3 


7.  (a;  y  +  1)3  14.  (a^  +  x  yf  21.  (a:  +  ^^  +  ^)' 


Composition, 

I.   Definitions  and  General    Principles. 

74.  A  quantity  composed  of  two  or  more  factors  other 
than  one  is  a  Composite  quantity. 

75.  A  quantity  composed  of  no  other  factors  than  itself 
and  one  is  a  Prime  quantity. 

76.  The  process  of  forming  a  composite   quantity  is 

Composition. 

Note. — Involution  is  a  special  kind  of  composition  in  which  the 
factors  used  are  all  alike. 

77.  Since  the  product  of  two  factors  with  like  signs  is 
positive  [P.  5],  and  with  unlike  signs  negative  [P.  6],  show 
that 

Brin,  35, — The  product  of  any  even  number  of  factors 
with  like  signs  is  positive. 

Prin.  36, — The  product  of  any  odd  ^lumber  of  factors 
with  like  signs  has  the  same  sign  as  the  factors. 

Prin,  37, — If  the  signs  of  an  even  number  of  factors  be 
chaiigedy  the  sign  of  their  product  will  remain  unchanged. 

Prin,  38, — If  the  signs  of  an  odd  number  of  factors  be 
changed,  the  sign  of  their  product  will  be  changed. 


68 


ELEMENTARY  ALGEBRA. 


SIGHT    EXERCISE. 

Name  the  signs  of  the  products  of  the  following  sets  of 
factors  : 

1.  (+«)x(+«)X(+«)X(+a) 

2.  (-  a)  X{-b)X  (-  c)  X  (-  d) 

3.  (+2)x(+3)x(+4)x(+5)X(+5) 

4.  (-  2)  X  (-  2)  X  (-  2)  X  (-  2)  X  (-  2) 

5.  (-  3)  X  (-  4)  X  (-  5)  X  (-  6)  X  (-  7)  X  (-  8) 

6.  (-  2)  X  (-  4)  X  (-  5)  X  (-  6)  X  (-  7)  X  (-  8) 

X(-9) 

7.  (-  a)  X  (-  J)  X  (-  c)  X(-d)X  (-  e)  X  (-/) 

8.  (-  x)  X{-y)X  (-  z)  X  (-  m)  X{-n)x  {- p) 

Tell  which  of  the  following  expressions  are  true,  which 
false,  and  why  : 

9,l{x-y)=-l{y-x)     11.  {z  -x)  =  -{x-  z) 

10.  —l(z  —  y)  =  l{y  —  z)      12.  2  (7i  —  m)  =  2  (7n  —  n) 

13.  xx{—y)x{-z)  —  xxyxz 


14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 


—  x)x{-y)Xz  —  xxyX{—z) 
y-x){z-y)  =  {x-y){y-  z) 

^-y){y-z){^-^)  =  (^-  y)  {y  -  ^)  {^  -  ^) 

n  —  m)  (q  —p)  {s  —  r)  =  (w?  —  n)  (p  —  q)  {r  —  s) 

V  —  z){z  —  v){v-\-  z)  =  {z  —  v)  (z  —  v)(z-}-  v) 

^-y){y-z){^-'^)  =  {^-  y)  (^  -y){u-  ^) 
^-y){y-^)(y-^)  =  {y-  ^f 

m  —  nY  (n  —  my  =z  (m  —  nY 

^  -  yf  {y  -  ^Y  (^  -  ^Y  (^  -  ^)'  =  (^  -  yY  (^  -  ^Y 


COMPOSITION.  69 

2.    Special   Principles  and  Applications. 

78.  The  sum  of  a  and  h  multiplied  by  their  difference, 
or  {a  -b)  {a-[-h)  =  a^-  h^.     Therefore, 

Prin.  39,— The  product  of  the  sum  and  difference  of 
two  quantities  equals  the  square  of  the  Urst  minus  the 
square  of  the  second. 

Applications. 

Ulustration.  —  Find    the    product    of    ^x^-\-^f    and 

27?-^  if. 

Solution :  (2a;«  +  3y»)  (2x«  -  3y»)  =  (2a:«)«  -  (3y»)«  [P.  39]  = 

EXERCISE    80. 

Find  the  value  of : 

1.  {x-\-y)(x-y)  4.  (3m  +  5/t)(3m-5w) 

2.  {a?-\-f)(^-y^)  5.  (4aa:  +  3^)(4aa;-3J) 

3.  {ax^\){ax-l)  6.  (23^y-{-dyz)(2x'y-3yz) 

7.  {o^y^-{-4:xy){x^y^  —  4:xy) 

8.  {5x'y-7z^)(63^y-{-7z^) 

9.  (x'y^z-'7)(x'y^z-{-'7) 

10.  (12  a:*  -  5  /)  {12a^-\-5  y^) 

11.  \{a  +  l)-^l\{(a  +  h)-l} 

12.  {{:>?-\-f)-Vz^\{{x'^f)-z^ 

13.  (a:  +  y)(.^-^)(^^  +  r) 

14.  (2a;  +  4)(2a;-4)(4a:2+16) 

15.  (3a:  +  52^)(3a;-5i/)(9a:2_^252^) 

16.  («2  r^  +  ^2  ^2)  (^2  ^.2  _  ^2  ^2)  (^4  ^T*  +  J*  «/*) 

17.  (a;  +  2)(a;~2)(:r  +  2)(x-2) 

18.  (2x-yY(2x^y){%x-^y) 

19.  {ax-\-byY{ax  —  b  yY 


70  ELEMENTARY  ALGEBRA. 

79. 


(1) 

(2) 

(3) 

X  +4 

a:  +4 

X  -4 

a;  +3 

X  -3 

a;  -3 

X^-\-A:X 

a;2  +  4a; 

a;2-4a; 

+  3a: 

+  12 

-3a:- 

-12 

-3a;  +  12 

x^^'lx 

+  12 

a;2+    a;- 

-12 

a?-tx-{-l% 

Notice. — 1.  That,  in  each  of  the  above  examples,  we  have  found  the 
product  of  two  binomials  having  a  like  term  {x),  and  two  unlike  terms 
(4  and  3),  the  latter  being  both  positive  in  (1),  one  positive  and  the 
other  negative  in  (2),  and  both  negative  in  (3). 

2.  That  the  first  term  of  each  product  is  the  square  of  the  like 
term ;  the  second  term  is  the  algebraic  sum  of  the  unlike  terms  times 
the  like  term ;  and  the  third  term  is  the  algebraic  product  of  the  un- 
like terms.     Therefore, 

Prin,  40, — The  product  of  tiuo  binomials  having  a 
common  term  equals  the  square  of  the  common  term,  and 
the  algebraic  sum  of  the  unlike  terms  times  the  common 
term,  and  the  algebraic  product  of  the  unlike  terms. 

Application. 

Illustration. — Find  the  product  of  a;^  +  8  and  x^  —  3 

Solution  :  {x"  +  8)  {x^  -  3)  =  {x^f  +  (8  -  3)  a;^  +  (8  x  -  3)  [P.  40]  = 

EXERCISE    40. 

Find  the  value  of  : 

1.  (a:  +  4)(a:  +  5)  9.  (2a;  + 4)  (2a;  + 3) 

2.  (a;+5)(a:  +  2)  10.  (3a:  +  3)(3.T+l) 

3.  (a:  +  5)(a:  +  6)  11.  (5  a:  + 2)  (5a;  +  3) 

4.  (a:  +  7)(a:  +  l)  i2,{x^2y){x-\-Zy) 
6.  (a:  +  8)(a:  +  3)  13.  {x^Q){x-b) 

6.  (a;+2)<a;  +  9)  •    14.  (a;  +  7)  (a;  -  3) 

7.  (a;  +  6)(a:  +  8)  15.  (a;  +  9)(a;-3) 

8.  {x  +  5)  {x  +  10)  16.  {x  -  7)  (a:  +  2) 


CROSS-MULTIPLICA  TION.  7 1 

17.  (a;-9)(a;  +  8)  23.  (3  a;  -  2y)  (3  a:  +  4  ?/) 

18.  {x  —  6)  (a;  +  12)  24.  (aa:  +  Z*)  (aa;  —  3  d) 

19.  {x  -  3)  (a;  -  7)  25.  (a;  +  «)  (a;  +  ^') 

20.  (a;  —  8)  (a:  -  G)  26.  (a;  -  «)  (a:  +  Z*) 

21.  {x  -  7)  (a;  -  10)  27.  {x  -\-a){x-  b) 

22.  (2  a;  +  3)  (2  a;  -  7)  28.  (a;  -  a)  (a;  -  b) 


80.  Cross-Multiplication. 


(1) 

(2) 

(3) 

2a;  -f   3 

2a;  -3 

2a;  -    3 

3a;  -f-  2 

3a;  +2 

3a;  -    2 

6a;2-{-    9a; 

Q7?-9X 

6a;2_    9a; 

+   4a; 

+  6 

+  4a;- 

-6 

—    4a;  +  6 

6a;2  +  13a; 

+  6 

Q7?  —  bX- 

-6 

6a;2_i3a;_^6 

Notice. — 1.  That  6  a;'  in  the  three  examples  is  the  product  of  the 
first  terms. 

2.  That  +  13  a;,  —5  a;,  and  —13  a;,  are  respectively  the  algebraic 
sum  of  the  products  obtained  by  a  cross-multiplication  of  the  first 
and  last  terras;  +  13a;  =  (3  x  3a;)  +  (2  x  2a;);  —  5a;  =  (— 3  x  8a;)  + 
(2  X  2a;);  and  -13a;  =  (-3  x3a;)  +  (-2  x  2a;). 

3.  That  +6,  —  6,  and  +  6  are  respectively  the  algebraic  products 
of  the  last  terms ;  +2x+3=  +  6;  +2  +  -3  =  —  6;and— 2x 
—  3  =  +  6.    Therefore, 

Prin,  4:1* — The  product  of  any  two  binomials  equals 
the  product  of  the  first  terfns,  and  the  algebraic  sum  of  the 
products  obtained  by  a  cross-multiplication  of  the  first  and 
second  terms ^  and  the  algebraic  product  of  the  second  terms. 

Application. 

lUustration. — Find  the  product  of  3  a;  —  5  and  2  a;  -j-  3. 
Solution:  (3a;  — 5) (2 a;  +  3)  =  3x  x  2a;  +  (+  3  x  3a;  — 5  x  2a;)  + 

(3  X  -  5)  [P.  41]  =  ea;"  -  a;  -  15. 


72  ELEMENTARY  ALGEBRA. 

EXERCISE    41. 

Find  the  value  of  : 

1.  (a;  +  3)(2a:+l)  12.  {^x -Zy)  {^x -2y) 

2.  (2 2: +  4)  (a; +  5)  13.  (m  +  2  7i)  (2m  -  3 7^) 

3.  («  +  2)(3fl^  +  4)  14.  (:i;2  +  5)(2a^-6) 

4.  (4«  +  3)(2«  +  2)  15.  (3a;2-4)(5iz;2_j_3) 

5.  (3g^  +  5)(2«  +  4)  16.  («a;  +  2)(2aic-5) 

6.  (a;+7)(5a,-  +  3)  11.  {4.x -^t)  {bx -^) 

7.  (2:2;+?/)(3:?:  +  2?/)  18.  {a?  -  y')  {^  a^  -  2  y"") 

8.  (3a^  +  25)(4a  +  ^>)  19.  (3a:2_2y)(4^_5^) 

9.  (2a;-3)(a;-5)  20.  (3a;  + 5)  (5^  -  3) 

10.  (4 a; -1)  (2  a; -7)  21.  (5icy  -  6)  (4:?;^  + 5) 

11.  (:?;-2«/)(3a;-5y)  22.  (3^^  + 5)  (Sm^  -  7) 


Exact  Division. 

Definitions. 

81.  Any  quantity  that  divides  a  given  quantity  without 
a  remainder  is  a  divisor  of  the  quantity. 

82.  All  the  different  quantities  that  divide  a  given  quan- 
tity without  a  remainder  are  the  divisors  of  the  quantity. 

Illustration. — The  divisors  of  aJ^h  are  1,  a,  a^,  b,  ah, 
and  a^J). 

83.  The  quantities  that  successively  divide  a  given  quan- 
tity and  the  resulting  quotients,  excepting  unity,  are  the 
continued  divisors  of  the  quantity.  They  are  the  same 
as  the  factors  of  the  quantity. 

84.  The  process  of  finding  one  or  more  divisors  of  a 
quantity  is  Exact  Division. 


EXACT  DIVISION, 


7a 


Special   Principles. 

85.  If  we  let  a  and  h  represent  any  two  quantities,  then 
will  a-\-h  represent  their  sum,  a  —  h  their  difference,  and 
a^  —  y^y  a^  —  d*,  a^  —  h^,  etc.,  differences  of  equal  even 
powers  of  them.     Now,  we  may  learn  by  actual  division, 

1.  That  a^—l^,  a^—b\  and  a^—b^,  are  divisible  hy  a  —  b. 

2.  That  a'—b^,  a^—¥,  and  a^—¥,  are  divisible  by  a  + J. 
Therefore, 

Prin,  42. — The  difference  of  the  equal  even  powers  of 
two  quantities  is  divisible  by  both  the  sum  and  the  difference 
of  the  quantities. 

86.  a*+^,  a*+J*,  and  a^-\-b^,  are  not  divisible  by  a-\-b. 
a*+^,  a*+*S  a^d  a^-^-b^,  are  not  divisible  hy  a—b. 

Therefore, 

Caution  1, — The  sum  of  the  equal  even  powers  of  two 
quantities  is  not  divisible  by  either  the  sum  or  the  difference 
of  the  quantities. 

SIGHT      EXERCI  SE. 

Tell  at  sight  which  of  the  following  examples  will  give 
rise  to  entire  quotients,  and  why  : 

1.  (a^-b')^(a-b)         il.  (4:a^ -9b^)  ^  (2a-3b) 


2.  (a*  -  b')  ^(a-^b) 

3.  {a'  -  ¥)  -r-(a-b) 

4.  (a«  -  b')  ^(a-^b) 

5.  {a'  +  b')^{a-b) 

6.  {a'-\-b^)-^(a^b) 

7.  (a«  -  b^) 

8.  (a'  -  ¥) 

9.  (a^  +  b") 


10.  {a^  -  b'') -i- {a^ -\- b^) 


(a^  -  ¥) 
(«3  +  b^) 
{a^  +  ¥) 


12.  (4^2  _  9  5,2)^  (2a  +  3  ^>) 

13.  (4«2-|-9J2)^(2a_3^>) 

14.  (16a*-81J*)-^(2a  +  3^>) 

15.  (16  a^  +  81  b^)  -^  (2  a  +  3  J) 

16.  (16a*-81J*)-^(2a--35) 

17.  (16a*  +  81**)-^(2a^-3J) 

18.  (a*2  _  J12)  ^  (^2  _  J2) 

19.  («12  _|_  J12)  ^  (^2  _  J2) 

20.  (flj'^  a^  —  Z>*  y^)  ^  (a  a;  —  6  y) 


74 


ELEMENTARY  ALGEBRA. 


87.  If  we  let  a  and  h  represent  any  two  quantities,  then 
will  a-^-l  represent  their  sum,  and  a^  +  h^y  w"  +  V',  a}  +  V, 
etc.,  sums  of  equal  odd  powers  of  them.  Now,  we  may- 
learn  by  actual  division  : 

That  a^  +  l^,  a^  +  h\  and  a?  +  IP  are  divisible  by  «  +  J. 
Therefore, 

Prifi.  d3* — The  sum  of  the  equal  odd  powers  of  two 
quantities  is  divisible  by  the  sum  of  the  quantities, 

88.  a^-{-b%  a^  +  b^  and  a'^-^b^  are  not  divisible  by 
a  —  b.     Therefore, 

Caution  2. — The  sum  of  the  equal  odd  powers  of  two 
quantities  is  not  divisible  by  the  difference  of  the  quantities, 

^       SIGHT      EXERCISE. 

Tell  at  sight  which  of  the  following  examples  will  give 
rise  to  entire  quotients,  and  why  : 

a'  +  b')  ^  {a^  -  ¥) 

a'  +  b')  -V-  {a'  +  *') 

a^-^b')-^{a^-b^) 

«3  _|_  J6)  -^  (^  -|_  ^2) 

a^^2'7b')-^{a  +  dP) 
a^-\-27b')^ia-3P) 
a'^-{-b'')-^{a  +  b) 

^12^J12)_j.(^2^^2) 
^12  4_J12)^(«4_^J4) 


1. 

(a'  +  x^)- 

-(a  +  2^)                  13.  ( 

2. 

(a'-^c^)- 

-  (05  —  :?r)                   14.  (< 

3. 

(a'  +  f)- 

-(a +  2/)                   15.  ( 

4. 

(x'-^f)- 

-(:?:-«/)                   16.  (( 

5. 

{8a^-{-27b^)-^{2a  +  Sb)     17.  (< 

6. 

{Sa'  +  P)---{2a-b)           18.  0 

7. 

{a'-\-d2b')-^{a  +  2b)         19.  (i 

8. 

(a^^b')^(a'-\-b')              20.  ( 

9. 

(l  +  ^-^)-(l  +  ^)                  21.  (i 

10. 

(2:^  +  l)-(^+l)                  22.  0 

11. 

(8  +  ^)^(2  +  :^-)                   23.  0 

12. 

(8a:3  +  27)-T-(2:r  +  3)           24.  (< 

25.  («^«^-32^'^«)-^(«2_^2^>2) 

26.  (8«^  + 

27b')^{2a^-{-3b') 

EXACT  DIVISION.  75 

89.  If  we  let  a  and  b  represent  any  two  quantities,  then 
will  a  —  b  represent  their  difference,  and  a^  —  P,  a^  —  b^, 
a'  —  i^  etc.,  differences  of  equal  odd  powers  of  them. 
Now,  by  actual  division  we  learn  : 

That  a^  —  b^,  a^  —  b^,  and  a?  —  b'^  are  divisible  by  a  —  J. 
Therefore, 

Prin,  44, — The  difference  of  the  equal  odd  powers  of 
two  quantities  is  divisible  by  the  difference  of  the  quantities, 

90.  a^  —  P,  a^  —  b^,  and  a"^  —  ¥  are  not  divisible  by 
a-\-b.     Therefore, 

Caitti&n  3, — The  differ ejice  of  the  equal  odd  powers  of 
two  quantities  is  not  divisible  by  the  sum  of  the  quantities, 

SIGHT      EXERCISE. 

Tell  at  sight  which  of  the  following  examples  will  give 
rise  to  entire  quotients,  and  why  : 

1.  {a^  -  X')  ^(a-x)  8.  (a^'  -  ¥')  ^  {a^  +  ¥) 

2.  {a'  -  ¥)  ^{a-\-b)  9.  (a''  -  ¥')  ^{a-b) 

3.  (a«  -  J^) -f- («2  -  ^2)  10.  (8a3_275^)-4-(2a-3*) 

4.  (a*  -  ¥)  -r  {a^  +  ¥)  11.  (8  a^  -  f)  --  (2  «  -  y^) 
6.  (a«  -  ¥)  -^  (a^  _  J3)  12.  {7?  -  27  /)  ■^{x'-^  y^) 

6.  (a^«  -  5^«)  -4-  (a^  -  ¥)         13.  (a:*  -  27  ^«)  ^  (rc^  +  3  y^) 

7.  (Sa:^  _  1)  ^  (2a;  -  1)         14.  (32  -  x"^)  -f-  (2  -  ar^) 

91.  By  actual  division  we  learn  that : 

(16a*  -  81  ^>*)  H- (2«  -  3  ^')  =  8a^  +  12a2  J  +  18«^ 

+  27  J' 
(16a*  -  81  ^'*) -4- (2a +  3^)  =  8a3  -  12a3  J  + 18«^r^ 

-27J» 
(a5-32J^«)     ^{a-'2¥)   =  a^ -\- 2  aH^ -^  4:  aH"^ 

+  8a^«  +  16J« 
(a'^  +  32J^«)     -^(a  +  2J2)   =  «i  _  2a3d2  _|_4a2^* 

-8a^>«  +  16i8 


76  ELEMENTARY  ALGEBRA, 

By  careful  inspection  we  may  observe  the  following 
laws  of  the  quotient : 

JPHn,  45. — 1.  The  number  of  terms  equals  the  expo- 
nent of  the  power  involved  in  the  terms  of  the  dividend, 

2,  The  terms  are  all  positive  when  the  divisor  is  the 
difference  of  two  quantities,  and  alternately  positive  and 
negative  when  it  is  the  sum, 

3,  The  first  term  is  found  by  dividing  the  first  term  of 
the  dividend  by  the  first  term  of  the  divisor. 

U.  Each  succeeding  term  may  be  found  by  dividing  the 
preceding  term  by  the  first  term  of  the  divisor,  and  multi- 
plying the  quotient  by  the  second  term  of  the  divisor,  dis- 
regarding the  signs, 

5,  The  last  term  may  also  be  found  by  dividing  the  last 
term  of  the  dividend  by  the  last  term  of  the  divisor. 

Note. — The  fifth  law  may  be  used  as  a  check  upon  the  fourth  to 
discover  errors  in  work. 

2.  Applications. 

EXERCISE    42. 

Tell  which  of  the  following  expressions  will  give  rise  to 
entire  quotients,  and  according  to  what  principle.  Write 
the  quotients  according  to  the  laws  in  [P.  45]. 

1.  {x^-y^)-^{x-y)  10.  (81a^-16)-^(3a;  +  2) 

2.  {x^  -  y^)  ^{x-y)  11.  («^  +  32)  -^  (a  -  2) 

3.  {x^  -  y')  ^  {x  -  y)  12.  {x^  -  y')  -^  (x  -  y) 

4.  {x'  -  y')  ^{x-  y)  13.  {x^  -  y')  -ir{x-\-y) 

5.  {x^  -  /)  -^(x-  /)  14.  (x^  -  y^)  -^{x^-  y^) 

6.  (82:^-27) --(22; -3)  15.  {^ -^  f)  ^  {^ -^ y") 

7.  (82:^-l)^(2a;  +  l)  16.  {x^ ^- y^) -^  {^^ -  y") 

8.  {a^  -  27  ¥)  ---{a-db)  17.  {x'  -Sy')^  {a^  -  2/) 

9.  (1  +  r^)  -^(1+x)  18.  (x'  -Sy')  -T-  {3^-i-2y^) 


FACTORING.  77 

19.  (32  ar^»  +  1)  -T-  (2  t>  +  1)  24.  {a}''  +  Z»^°)  -f-  (a*^  +  h') 

20.  (a^2  ^  J12)  ^  (^3  _  j3)  25.  (a^*^  +  Z'^")  ^  {a"  +  Z>^«) 

21.  (1  +  729  x^)  -^  (1  +  9  o.-^)  26.  (««  -  729  /)  -^{a-^y) 

22.  (64  -  ««)  -^  (4  +  ^2)  27.  (a«  -  729  /')  -^  (a  +  3  ^) 

23.  (625  a*  -  1)  ^  (5  a  +  1)  28.  (a«  +  729  y^)  ^  (a  +  3  y) 

29.  (16a*-81^*)-^(2«  +  3^') 

30.  (x«  +  64 1/'')  -^  (a:^  _j_  4  ^2) 

31.  (8a:«  +  27/)-^(2a;2^3y3) 

32.  (a;^<>  +  32  y"^)  -^  (a:«  +  2  /) 

33.  {7?y^  — 7^\f')-^{x^y  — X %f) 

34.  (a3a;»  +  J3^«)-^(aa:5_^j^2) 

35.  (256  7^  +  10,000)  -^  (4  a  +  10) 

36.  (a«  +  729  y«)  -T-  (««  4-  9  ^/2) 

37.  (512a^^^  +  c3)-^(8a5  +  c) 

38.  (8a;«  +  273^*)-^(2  2:2_^3^3) 

39.  (a;i«  +  32  /°)  -^  (a:^  +  2  y*) 


Factoring. 

I.   Definitions  and   Principles. 

92.  The   quantities  multiplied  together  to  produce  a 
given  quantity  are  the  Factors  of  the  quantity. 

93.  The  prime  quantities  multiplied  together  to  produce 
a  given  quantity  are  the  Prime  Factors  of  the  quantity. 

94.  A  composite  quantity  may  have  two  or  more  sets  of 
factors,  but  it  can  have  only  one  set  of  prime  factors. 

Thus,  a^b^  =  ar  X  lr  =  arh  x  h  =  ab^  X  a  =  ab  X  ab  = 
axaxb"  =  a"xbxb  =  abxaxb  =  axaxbxb. 
The  last  is  the  only  set  of  prime  factors. 


78  ELEMENTARY  ALGEBRA. 

'  95.  The  process  of  finding  the  factors  of  a  quantity  is 
Factoring. 

96.  ah  -^  a  =  l) ',  but,  a  and  h  are  the  factors  oi  ah-, 
therefore, 

JPrin,  46. — A  divisor  of  a  quantity  is  one  of  the  two 

factors  of  the  quantity/,  and  the  quotient  is  the  other. 

97.  Since  a  diyisor  of  every  term  of  a  quantity  is  a 
divisor  of  the  quantity  [P.  15],  and  a  divisor  of  a  quantity 
is  a  factor  of  the  quantity  [P.  46],  it  follows  that, 

JPrin,  47. — A  factor  of  every  term  of  a  quantity  is  a 
factor  of  the  quantity. 

2.  Problems. 

1.  To  factor  a  polynomial  having  a  common  factor  in 
its  terms. 

Illustration. — Factor  a^c^  —  ac^ -\-a^c. 

Fornii 

{a^(^  —  ac^-\-a^c)  =  ac{ac  —  c-\-a) 

Solution :  We  see  by  inspection  that  a  c  is  a  factor  of  each  term  of 
the  polynomial;  it  is  therefore  a  factor  of  the  polynomial  [P.  47]. 
Dividing  by  a  c,  the  quotient  ac  —  c  +  a  is  the  other  factor  [P.  46]. 
Therefore,  (a^c^  —  ac^  +  a^c)  =  ac{ac  —  c  +  a). 

EXERCISE    43. 

Factor : 

1.  a^-^ab  7.  Qx^y^  -  12a^y^  -  18xy 

2.  ah-hc  8.  10a^  +  152;3-20a;2 

3.  Qi?-\-axy  9.  7 r^  -  14 r^  +  21  r* 

4.  a;3  _|_  3  ^  _  2  a;  jq.  2  «  (a  +  &)  +  3  J  («  +  5) 

5.  3  a^  —  6  a  5  +  9  a  5-'  li.  a{a  —  x) —  l{a  —  x) 

6.  2a^x-{-4:d^a^  —  6a''z'^     12.  c  (m  -\- n) -\- d  {m -{■  n) 
13.  12  aH^  c^  -  Ua^bU''  -^dQa"  I^  c 


FACTORING.  79 


14.  10/  q^  +  \bp^  f-'^Opqr 

15.  24a:8«^«- 36  2:5^9^43^^10 

16.  ^c{a'^h^)-^d{ar-\-h^) 


2.  To  factor  the  difference  of  two  squares. 
Illustrations. — Factor  a^  —  W,  7^  —  ^,  and  7^  y^  —  oc^y^. 
Solutions :  1.  a^  -  V^  =  {a -\- h)  {a  -  b)  [P.  39 j. 

2.  a;*-y*  =  (a:«  +  y')(x^-y')  [P.  30]  =  (a:»  +  y'){x  +  y) 

(a: -2/)  [P.  30]. 
^.  7fiy^-Qi^y^  =  a^y'^{x^-  y^)  [P.  47  and  46]  =  x^y^ 

{x  +  y){x-y)[¥.m\, 

EXERCISE    44. 

Factor : 

1.  a^  -  4^>2  11.  «*  -  ;2*  21.  {a  -\-hY-c^ 

2.  4a2  -  25  ^^2  12.  ««  -  5*  22.  (a  -  xf  -  if 

3.  9a:2_49^2  13.  IGa*  -  81;2*  23.  (m  -  7z)2  -  1 
4.a2«^2_4  14.  81  /  -  256  ;z*  24.^-{x  +  ijf 

5.  16  -  ^8  15.  a:*/  _  a;2^  25.  c^  -  (a  +  ^>)2 

6.  ic^  -  64  16.  a;«  -  /  26.  c^  -  (a  -  ^)2 
l.^if-  100        17.  625  -  2^                 27.  25  a^-(x-  yf 

8.  81  -  JZ^  18.  a^-f  28.  16  -  (;z  -  a:)^ 

9.  a^W(?-  36       19.  a;i2  _  ^^  29.  1  -  (a;  -  yf 
10.  x'y^-y^  z^       20.  m^  -  ?i^^  30.  49  -  4  (a;  +  yf 


3.  To  factor  the  sum  or  difference  of  the  equal  odd  powers 
of  two  quantities. 

niustration.— Factor  c^  —  l^,  a^  +  W,  and  a^  —  W', 
Solutions :  \.  a^  -h^  =  {a-h){a>  ^  ah  ^V^)  [P.  44  and  45]. 

2.  a'  +  fe»  =  (a  +  &)(a«  +  a 6  +  6«)  [P.  43  and  45]. 

8.  a«  -  6»  =  (a»  +  &»)  (a*  -  5»)  [P.  30]  =  («  +  &) 

(a«  -  a6  +  6»)(a-  6)(a«  +  a&  +  6')  [P.  43,  44,  45J. 


80  ELEMENTARY  ALGEBRA, 

EXERCISE    43. 

Factor  the  following  binomials.      Three  can   not  be 
factored  : 

1.  7?  —  y^  11.  a^-\-l  21.  a?-\-if 

2.  o^-\-y^  12.  a^  —  1  22.  x^  —  y* 

3.  a?  —  \  13.  x^  +  8  23.  a^-{-x 

4.  a^-\-l  14.  a;«  —  8  24.  2;5  —  Sir* 

5.  a;^  —  8  16.  x^  +  3/5  25.  a;^  i/^  +  2^ 

6.  ar^  +  8  16.  x^  —  y^  26.  m^  +  7i' 

7.  8  fl^=^  + 1^  17.  a;^  +  ^^  27.  64  m«  +  125  n^ 

8.  8a^  — ^^  18.  32a;5  — 2/^*  28.  3^y^  —  xy 

9.  27  a^- 8  P  19.  16  x^  +  81  y^  29.  a;^«  +  a;  ^/^ 
10.  27a^  +  8^^  20.  8a;«-27/  30.  (a^  +  ^j^-^^^ 

31.  (x  +  i/)3  +  z^  33.  a:^  +  (^  +  ;2)' 

32.  X^-{y  +  zy  34.   («  +  J)3_(c  +  c^)3 


4.  To  factor  a  trinomial  that  is  a  perfect  square. 
Ulustrations. — 

1.  a^  +  2ab-\-b^  =  {a  +  h){a-]-b),  since  {a-]-by  = 

a^-\-2ah-^h^  [P.  31]. 

2.  fl2  _  2  ff  J  -f  Z>2  _  (^  __  J)  (^  _  j)^  since  (a  -  If  = 

a2_2fl^J  +  Z/2  [P.  32]. 

3.  4a2  +  12a^5  +  9J2=(2«  +  3^)(2a  +  3^),  since 

{2a-^^l)f  =  ^aJ'-\-12ab-\-^h^  [P.  31]. 
4  4fl2-i2a$  +  9Z'^=(2«-3Z')(2a-3^),  since 

(2«-3^f  =  4a2_i2«J  +  9^2  [P.  32]. 

98.  A  trinomial  is  a  perfect  square  when  two  of  its 
terms  are  perfect  squares,  and  the  other  term  is  ±  twice 
the  square  root  of  their  product. 


FACTORING.  81 

EXERCISE    46. 

Factor  the  following  trinomials.     Three  can  not  be 
factored  : 

1.  a?  +  'ilxy-\-y^  13.  IQx" -nx'y'' ^^ly^ 

2.  T^-'ixz-irz'^  14.  ar*  +  22^2  +  4 

3.  x'^2x-{-l  15.  x^''-\-%o^-\-l 

4.  a;2-4a;  +  4  16.  T^y"" ■\-4.x''y^ ^^.x" y^ 
6.  a:2_|_i8a;  +  81  17.  a:^  -  2 a:* 2^  +  3^ 

6.  4a;8_i2a;  +  9  18.  4ir*  + 14a:2_|_49 

7.  9a:2_^i2a;y  +  4«^  19.  4 a:«  +  12 a:^  _^  9 

8.  25a:2_pioa;  +  l  20.  9:r«  -  36ar*^2_|_  35^^ 

9.  a;*-12a^  +  36  21.  lOOa^  -  110  a;  y  + 121  y* 

10.  Q^-%j?^lQ  22.  a;*^2_^2a:2y3^_^^^2 

11.  a:«  +  2a:3^^yj  23.  a;«  +  42^6  -  4a:3^ 

12.  a2j2_2«Jcc?  +  c2j2         24.  16  2:8  +  256/ -128  a:*/ 


6.  To  fjEUjtor  a  trinomial  that  io  the  product  of  two 
binomials  having  a  like  term. 

Illnstratioiis. — 

1.  a^  4_  3  05  4.  2  =  (rt  +  2)  (ff  +  1),  since  (a  +  2)  ( «  +  1) 

=  a2  +  3«  +  2  [P.  40]. 

2.  a^  _  5  flf  _j_  6  =  («  -  2)  (a  -  3),  since  {a  -2)  {a-  3) 

=  a''-6a  +  6  [P.  40]. 

3.  a^  +  2  rt  -  8  =  (a  +  4)  (a  -  2),  since  («  +  4)  (a  --  2) 

=  a^-\-2a-S  [P.  40]. 

4.  (4a2-.4fl5_i5)  =  (2a  +  3)(2a-5),  since 

(2a  +  3)(2a-5)  =  4a2-4«-15  [P.  40]. 

99.  A  trinomial  is  the  product  of  two  binomials  having 
a  like  term,  when  the  first  term  is  a  square,  and  the  last 
term  is  the  algebraic  product  of  two  factors  whose  sum. 


82  ELEMENTARY  ALGEBRA. 

multiplied  by  the  square  root  of  the  first  term,  will  give 
the  middle  term.  The  square  root  of  the  first  term  is  the 
like  term  of  the  binomials,  and  the  factors  of  the  third 
term  are  the  two  unlike  terms. 

EXERCISE    47. 

Factor  the  following  trinomials.      Three  can  not  be 
factored  : 

1.  ic-  +  8a;  +  15  13.  4.x^-\-ix^^ 

2.  x^-\-5x  +  4c  14.  4a^-\-Ux-{-12 

3.  a^-\-6x-{-8  15.  9a^+9x-\-2 

4.  a^-7a-\-12  16.  x^ -{- 4: a x -}- 3 a^ 

5.  a^-9a  +  U  17.  x^-2ax-15a^ 

6.  «2_i3^4-40  18.  4:X^  —  Sax  —  3a^ 

7.  a;2_^2a;-15  19.  9  y^ -\-3y  z  -  2z^ 

8.  x^-^3x-28  20.  dea^-\-24:bx-6i^ 

9.  a^-i-6x-16  21.  4:a^a^-4:ax- 16 

10.  x^-4:X-6  22.  x^-i-Hx^-12 

11.  x^-4=x-21  23.  a^-7aa^-\-12a^ 

12.  a^  —  2x  —  S0  24.  4:a^  +  Sa^  —  3 


6.  To  factor  a  trinomial  that  is  the  product  of  any  two 
binomial  factors. 
Illustrations. — 

1.  2x^-{-6x-{-2  =  {x-\-2){2x-]-l),  since 

(x  +  2)(2x  +  l)  =  2x^-{-6x-{-2  [P.  41]. 

2.  6a^  —  13x-{-6  =  {2x-3){3x  —  2),  since 

{2x-3){3x-2)  =  Qa^-13x-]-6  [P.  41]. 

3.  2x^-{-x  —  lb  =  {x-{-3){2x-5),  since 

{x-]-3){2x-5)  =  2a^-\-x-U  [P.  41]. 

4.  6  a;2  —  11 2:  -  10  =  (2  a:  —  5)  (3  a;  +  2),  since 

(2x-5){3x-\-2)  =  6x^-nx-10  [P.  41]. 


FACTORING.  83 

100.  The  first  terms  of  the  factors  are  the  factors  of  the 
first  term  of  the  trinomial,  the  last  terms  of  the  factors  are 
the  factors  of  the  last  term  of  the  trinomial,  and  the  last 
terms  of  the  factors  are  so  arranged  with  the  first  terms 
that  the  algebraic  sum  of  the  products  obtained  by  multi- 
plying the  first  term  of  each  factor  by  the  second  term  of 
the  other  will  give  the  middle  term  of  the  trinomial. 

Note. — This  and  the  following  problem  may  be  omitted  until  the 
class  reaches  page  192,  if  desirable. 

EXERCISE    48. 

Factor  the  following  trinomials.  Two  can  not  be  fac- 
tored.    Why  ? 

1.  2a;2_j_5^_^3  jq.  ISa^-f  20a-35 

2.  2  2:2  _|_  11  ^  _|_  12  11.  2  ic2  _|_  19  ^  _  35 

3.  6ar  +  7a:  +  2  12.  2a^-\-ab-h^ 

4.  Qa?-\-llx-{-^  13.  2a2-|-5aJ_|-2J2 

5.  2x'-'tx-\-Q  14.  Qx^-llxy-^^y^ 

6.  2 a,-2  +  .T  —  6  lb.  Qu^-\-buv  —  Qv^ 
n.  2x'-x-15  16.  23^-7xy-dy^ 

8.  12a:2^iia._i5  ^^^  6a^-lla^-d6 

9.  6a^-{-a-16  18.  2a^b^  +  ab-Q 


7.   To  factor  a  trinomial  that  is  the  product  of  two  ttino- 
xnials  of  the  form  of  x^  +  xy  +  y^  and  x^  —  xy  +  y^. 

Solution  :  The  product  of  x^  +  xy  +  y^  and  x^  —  xy  +  y^  is 
a^  +  x*y^  +  y* ;  therefore,  a  trinomial  is  the  product  of  two  trinomials 
of  the  form  of  x*  +  xy  +  y^  and  x^  —  xy  +  y^  when  all  its  terms  are 
positive  squares  and  the  middle  term  is  the  square  root  of  the  product 
of  the  other  two. 

Rule. —  The  factors  may  be  obtained  by  extracting  the 
square  root  of  each  term  and  making  the  middle  term  of 
one  factor  positive  and  that  of  the  other  negative. 


84  ELEMENTARY  ALGEBRA. 

Illustrations. — 

1.    a*4-«2  4-l  =  («2-|-^_|-l)(a2_«-|-l). 
EXERCISE    49. 

Factor  the  following  trinomials.     One  can  not  be  fac- 
tored : 

1.  a;*  +  a:2_|.i  10,  %lx^  +  mQ?y^^Uf 

2.  a;*  +  4a;2  +  16  ii.  a^ -\- a^ y^ -\- y"^ 


,16 


3.  a*  +  a2^>2_|_^4  12^  ttH-^  +  a^ic^^g.^^ 

4.  a^-^aW-\-(^  13.  a'^y^-^a^y^-\-a^f 

5.  16a*  +  4a2  +  l  14.  a8&8_^4a*5*  +  16 

6.  a8  +  4«4j2  +  i654  15.  81  +  9^2 _j_j4 

7.  a^  +  a;*/  +  /  16.  625  +  25 a.-^ /  +  a;* / 

8.  a:8  +  iC*«/^  +  «/^3  17,  256  +  16;2*  +  ^« 

9.  ic^  — a^^  +  /  18.  a;*/  +  a:2^2;2_^/i2* 


Ulnstrations.- 

8.  To  factor  polynomials. 

1. 

Factor  ax 

'-{-hy-hx-ay. 

SoltLtion : 

ax  +  hy  —  hx  —  ay  — 

ax  —  hx  —  ay  +  &2/  = 
x{a  —  h)  —  y  {a  —  h)  = 
{a-b){x-y)  [P.  46,  47]. 

2. 

Factor  a? 

-2a;  2/  +  /-;^^ 

SoltLtion : 

ic^  — 2a:y  +  2/2  — £2  = 
(x^-2xy  +  y^)-'Z^  = 
(x-yf-z^  = 
{x-y-z){x-y  +  z)  [P.  39]. 

3. 

Factor  x^ 

-2/'-2«^2;-;z2, 

Solution : 

x^  —  y^  —  2yz  —  z^  = 

x^-(y^  +  2yz  +  z^)  = 

x'-^(y  +  zf  = 

{x  +  y  +  z){x-y  +  z)  [P.  39]  = 

(x  +  y  +  z){x-y-z)  [P.  23,  24]. 

FACTORING,  85 

EXERCISE    80. 

Factor : 

1.  ax-\-ay  +  hx-\-ly  ii.  or -\-^ah-\-lr  -  c^ 

2.hx  —  hy-\-cx  —  cy  12.  a?  —  ^x-\-l—  y^ 

3.  ax  — az  —  bx-{-bz  13.  a^  —  y^  —  2y  —  1 

4.  ab  +  2i-\-3a-\-Q  14.  4:0^ +  4:xy -\-y^  -  z^ 

5.  9-^3x  —  dy  —  xy  15.  4:Z^  —  4:0^  —  4:X  —  1 

6.  2ax-i-3ay-{-4:bx-^6by  16.  a^-\-2ab-{-b^ -16 

7.  6ax-\-4:ay—dbx  —  eby  17.  25— a^  —  2«a;  —  a^ 

8.  aba^'j-2ax-\-3bX'i-6  18.  a^  —  a? -\-2x —  I 

9.  «a;y4-6a  — Ja;y  — 6J  19.  a:*  +  2a:2^2_^^  _  ^2 
10.  a^  7?  ^  a^  y^  -V^  Qi?-W  y^  20.  m*  -j9*  -2fq-  q^ 


Miscellaneous  Examples. 

EXERCISE    81. 

(Take  out  monomial  factors  first.) 
Factor : 

1.  a^b  —  ab^  12.  ^a7^  —  6a3^f-{-3axy^ 

2.  3fl3  — 12a  13.  4ty'^-i-Sy*  +  4:y 

3.  2a^-2ab^  14.  2  a:^  _^  iq  ic^  + 12  a; 

4.  3a35-f3J4  16,  a:3^_92^y_^20rz;«^ 
6.x^y  —  xy^  16.  4: a^b-{- 4: a^b  —  lGSab 

7.  d}b-aV  18.  a»  +  a2(&  +  c)^ 

8.  2a»J2_^2a2J8  19,  a^^  -  a^c^  -  2ac3 

9.  6a«  +  10ay  +  5i/2  20.  4a5  +  8fl  +  12 J  +  24 

10.  2a3c  +  12a2c  +  18ac        21.  axy  —  bxy—ay^-{-by^ 

11.  7? y^z  —  2oc^yz-\-xz  22.  a^  —  y^  —  x-{- y 


86  ELEMENTARY  ALGEBRA. 


23. 

(xJryf-x-y 

36. 

«2  +  2  a  +  1  -  Z»2 

24. 

{a^hfx^-c^x^ 

37. 

aH^-a^-2ah- 

25. 

^-%x''y-^xy'-x%'' 

38. 

\-x^-%xy-y^ 

26. 

a^-^x^y-]-^xy^-y^ 

39. 

a^  —  m^  -\-2mn  — 

27.  a:*  +  6  a;3  +  12  2;2  +  8  ic         40.  1  -  (a  +  ^')^ 

28.  a*/  -  1  41.  a^  -\-aH^-^  h^ 

29.  a?  y^  —  z^  42.  {x  +  «/)^  —  (ic  —  yY 

30.  a;i2^y2  43^  a»  +  «8  2/7 

31.  x"^  -  y^^  44.  (a  +  2')2  -  2  («  +  J)  +  1 

32.  TTv'n^  —  m  45.  a;^  —  (a;  +  i/  +  ;2;)^ 

33.  121  «^  + 144  ¥  +  264  a^  J2    45,  a;*  -  (2/  +  ;^)* 

34.  16  a^  -f  8  tt  ^  -  3  ^2  4n.  7?-\-^x^y-\-^xy^^y^ 

35.  3  m  7^  —  a  m  ?^  +  2  a  —  6      48.  1  —  («  —  ^)^ 

49.  aHc-^ahH^aHd-^-aTy'd 

50.  3a^  — 15a:?/  — 2^a;+10J^ 


Highest  Common  Divisor. 

I.   Definitions  and   Principles. 

101.  A  divisor  of  each  of  two  or  more  quantities  is  a 
Common  Divisor  of  the  quantities. 

102.  The  common  divisor  that  contains  the  greatest 
number  of  prime  factors  is  the  Highest  Common  Divisor. 

103.  Quantities  that  have  no  common  divisor  except 
one  are  prime  to  each  other. 

104.  Since  a  quantity  equals  the  product  of  its  prime 
factors,  it  is  divisible  by  the  product  of  any  two  or  more 
of  them ;  hence,  too,  each  of  two  or  more  quantities  is 


HIGHEST  COMMON  DIVISOR.  87 

divisible  by  the  product  of  any  two  or  more  of  their  com- 
mon prime  factors  ;  and  therefore, 

Prin,  48. — The  highest  common  divisor  is  the  product 
of  all  the  common  prime  factors, 

106.  The  abbreviation  H.  C.  D.  stands  for  highest  com- 
mon divisor. 

2.   Problems. 
1.  To  find  the  highest  common  divisor  of  monomials. 

niustration.— Find  the  H.  C.  D.  of  Ua^l^d",  Ua^l^(P, 
and  ^^Qa^'Pd'K 

Solution:  Ua'^Pd'  =2  X  2  X  2  X  2  X  a^  X  h^  X  C*' 
24a3J2c2  =2x2x2xSXa^Xb^X(^ 
^QaH^d^  =  2  X2x3x3x«*XJ'Xrf2 
.-.    H;0.  D.  =2x2Xa^X*^  =  4«2  52[p.  48J. 

Another  Solution :  The  H.  C.  D.  of  the  coefficients  is  4 ;  of  the 
o's  is  a' ;  of  the  6's  is  6* ;  and  c  and  d  are  not  common  to  the  three 
quantities.    Therefore,  the  H.  C.  D.  is  4  a'  6*. 

Mtde. — Find  the  highest  common  divisor  of  the  numeri- 
cal coefficients,  annex  to  it  the  different  common  literal  fac- 
tors, giving  each  the  lowest  exponent  it  has  in  any  one  of 
the  quantities, 

EXERCISE    82. 

Find  the  H.  0.  D.  of : 

1.  ^x^y,  12xy^,  and  24a;2^2 

2.  15aH^c^,  25  a- b^  and  30^2^2 

3.  20xy^z,  SOs^yz,  and  AOxyz^ 

4.  20aH\  26an^c,  and  S5aH^(^ 

6.  IS  m^n\  24:am,^n^  and  d6bm^n^ 

6.  {a-\-by,  (a-i-by,  and  {a -{-by 

7.  3{x-{-yy,  6{x-\-yy,  and  9(x-{-yy 

8.  m{m-\-  ny,  m^  (m  +  ny,  and  m^  {m  -j-  n) 


88  ELEMENTARY  ALGEBRA, 

9.  2ax{x^-\-y%  4.ax^x'^y^)\  and  ^ a^ a? {a? -\- y-f 

10.  4  (m  —  ny,  6{n  —  rnf,  and  8  (m  —  vif 

11.  xy{a  —  if,  o?y(a  —  lY,  and  if{a  —  bf 

12.  (fl^  +  ^')  (a  -  ^),  {a  +  5)2  (^  -  hf,  and  (a  +  5)2  {a  -  hf 

13.  2  (rr  -  yf,  4  (a;  -  2/)^  6  (a;  -  ^)S  and  ^{x-  yf 

14.  3a(m  — 7e)^  Qah{m  —  nf,  9a^(m  —  n)\  and 

12  aW  {m  —  nf 

15.  (m  +  ?i)  (m  —  7i),  (m  -\-n){n  —  m),  and 

(m  +  nf  (m  -  nf 

16.  {p  -q){a-  h),  {q  - p)  {a  -  I?),  and  {p  -  q)  {i  -  a) 

17.  {a  -  J))\  (b  -  of,  and  (a  -l)){h-  a) 

18.  ah{a  —  V),  —l){h  —  a),  and  a  {a  —  hf 


2.  To  find  the  highest  common  divisor  of  polynomials. 

ninstration.— Find  the  H.  C.  J),  ot  xf^  —  f,  x^y  —  xf, 
and  Q?  —  7?  y  —  X  y^  -\-  y^. 

Form. 

^-t^i.x^-f)i.^^^t)  =  {x-y){x^y){x^-^f) 

7^ y  -  xy^  =z  xy  {x^  -  y^)  =  xy  {x  -  y)  {x  -\-  y) 
a?  —  x^y  —  xy^-\-y'^  =:a?{x  —  y)  —  y^{x  —  y)=^ 

(^  -  y^)  i^-y)  =  (^  +  y)  (^  -y)(^-  y) 

H.  Q.T>.  =  {x^y){x-y)  =  x'- /  [P.  48]. 

Solution :  We  resolve  the  quantities  into  their  prime  factors,  and 
observe  that  a  +  5  and  a  —  h  are  the  only  common  factors ;  therefore, 
{X  +  y){x-  y\  or  x^  -  y\  is  the  H.  C.  D.  [P.  48]. 

EXERCISE    63. 

Find  the  H.  C.  D.  of : 

\.  a-{-h  and  a/^  —  h^  3.  {a  —  hf  and  a^  —  W 

2.  (a  +  hf  and  a*  -  5*  4.  (a  +  hf  and  a^  +  5^ 

5.  x^  —  xy,  x^  —  y^,  and  x^  —  2xy -\-y^ 


HIGHEST  COMMON  DIVISOR,  89 

6.  {x  -{-  yY,  7?  —  y^,  and  o?  -{-xy 

7.  or  —  y^,  7?  —  1xy-\-y'^,  and  x^y  ^x^ 

8.  {a;  +  y)^  a?-\-7?y,  and  2:^4-^^ 

9.  a;2_42:,  x^-^x  +  lQ,  and  a;2_2a;-8 

10.  «*  +  4a3  +  4a2,  a35-4«^,  m^a^b-^ba^h-^Qa^l) 

11.  a,*^  —  8 a:^  +  15 a;,  ic^y  —  8a;^  + 15?/,  and 

x^z  —  ^xz-\-lbz 

12.  3a-2-3/,  cc*- 2  2:2^2^^^  a^jj  x'y-f 

13.  a:^  +  a:  y  4-  ^  ^>  ^  V  -{- V^  -\- V  ^i  ^^^  ^  —  (^  +  ^)^ 

14.  a:2  +  a;_6,  ar  +  7a;4-12,  and  x'-^x-lb 

15.  a:^  +  27,  a^  +  5  a;  +  6,  and  a:^  _|_  g  ^  _^  9 

16.  3? -\-ax-\-hx-\-al)  and  3?  -{-ax-\- cx-{-ac 

17.  vf?  —  v^,  am-{-an-\-lm-\-ln,  and  w^  +  2 ?/i 7i  +  w* 

18.  7^-\-xy^,  x^  -^%7?y^  -\-y^,  and  aa:^  +  «y^ 

19.  a:*  +  4 a:^ y  +  4 3/2,  a^  — 4^^,  and  a:^y  +  2a;^* 

20.  7?  —{y-\-  zY,  y^  —  (x-{-  zf,  and  z^  ^{x-{-  yf 

21.  a;*  -  (2/  +  2)«,  y^-{x^  2)^  and  4  -  (a;  +  3/)^ 

22.  a;^^-y^  ^y +  2«*y*  +  a;y^  and 

23.  3^y-\'12x^y-}-Z5xy,  ic^ -f- 3  a:*  -  28 a:^^  and 

a:^;2  — a:^;2!  — 56a?jj; 

24.  6a:2 _|_i0a;-24,  2a;2_2a;_24,  and  8a;«  +  22a;-6 

25.  x^-y^  a^  +  a:2«/^  +  /,  and  a:^  y  +  3^5  ^^2  _p  3.  ^ 

26.  a^  +  5x^-]-6x,  a^y-^z^y-exy,  x^-x^-12a^ 

27.  a:*-16,  ai^-^Sx^  +  16,  a:*4-2a:2_8 

28.  a^'-a',  x^'-2a^o?-\-a?',  x' -  a7?  -  a'x-^-a^ 

29.  x^  —  f,  ^-\-^y  +  y%  x*-\-3^y^-\-y* 

For  highest  common  divisor  by  successive  division,  see  Appendix. 


90  ELEMENTARY  ALGEBRA. 

The  Lowest  Common  Multiple. 

I.   Definitions. 

106.  A  quantity  that  exactly  contains  a  given  quantity 
is  a  Multiple  of  the  quantity. 

107.  A  quantity  that  exactly  contains  each  of  two  or 
more  given  quantities  is  a  Common  Multiple  of  those 
quantities. 

108.  The  common  multiple  that  contains  the  least  num- 
ber of  prime  factors  is  the  Lowest  Common.  Multiple, 

109.  The  abbreviation  L.  C.  M.  stands  for  lowest  com- 
mon multiple. 

110.  The  L.  0.  M.  of  aH^c,  aW(?,  and  aHH^  must 
contain  each  of  these  quantities  [Art.  107]  : 

ah^c^  =aXhXhxhXcXc 
a^h^c^=zaXaXhXhXcXcXc 

To  contain  a^  W  c,  the  L.  0.  M.  must  contain  the  prime 
factors  a,  a,  a,  h,  h,  c. 

To  contain  a  ¥  c^,  it  must  contain  the  additional  prime 
factors  l  and  c. 

To  contain  a^l)^(^,  it  must  contain  the  still  additional 
prime  factor  c. 

Since  these  are  the  only  factors  required  to  contain  each 
of  the  quantities,  and  all  are  necessary,  the 

lj.G.lA.  =  aXaXaXhXiXhXcXcXc  =  a^b^c^, 
Therefore, 

Trin.  49. — The  loioest  common  multiple  of  two  or  more 
quantities  equals  the  product  of  all  their  different  prime 
factors,  each  taken  the  greatest  number  of  times  it  occurs 
in  any  one  of  them. 


LOWEST  COMMON  MULTIPLE.  91 

2.   Problems. 

I.  To  find  the  lowest  common  multiple  of  monomials. 

niustration.— Find  the  L.  C.  M.  of  UaHH,  3Ga^JV, 
and  bQaH^c^, 

Solution '.  %4:a^h^c  =2x2x2xSXa^Xb^Xc 
3eaH^(^  =  2x  2  X  3  X3  X«^  X  &2><^3 
56a2^V  =  3x2x2x  7  X  a^  X  ^'*  X  c^ 

,-.  L.  CM.  =2X2X2X3  X3X7X«'^X^'XC3  = 

504:a'b^(^  [P.  49J. 

Another  Solution :  The  L.  C.  M.  of  the  coefficients  is  504 ;  of  the 
a's  is  a* ;  of  the  6's  is  b* ;  and  of  the  c's  is  c\  Therefore,  the  L.  C.  M. 
of  the  quantities  is  504  a'  6^  c^. 

Utile, — Find  the  least  common  multiple  of  the  numeri- 
cal coefficients ;  annex  to  it  all  the  different  literal  factors 
found  in  the  quantities,  giving  each  the  highest  exponent  it 
contains  in  the  quantities. 

EXERCISE    84. 

Find  the  L.  C.  M.  of : 

1.  lOx^y,  15xy\  and  20a^y^ 

2.  na^b\  ISai^c,  and  24:a'^c^ 

3.  24:aa^,  d2bxy,  and  48c/ 

4.  22a:*/,  Sdx'z^,  and  Uy^z"" 

5.  2Aab^a^,  36a^x^z,  and  ^Sb^z^ 

6.  48m^w^  bem^nx,  and  63^^0:3 

7.  (a-\-by,  (a  +  J)^  and  {a-\-by 

8.  25{x  +  yY,  60{x-i-y)^  and  100(x-\-yy 

9.  a^  (x  —  y),  a^  {x  —  yf,  and  a*  {x  —  yY 

10.  8 a* (a; +  ;?)*,  12 ah {x-^-zf,  and  24.b^x  +  zy 

II.  Qa^^^iT'-i-y^y,  lSx^{a^-{-y^y,  and  36a^z(a^-}-y^y 
12.  {a-{-b){a-b),  {a^  +  b^){a-{-b),  and  (a"" -\- b')  (a  -  b) 


92  ELEMENTARY  ALGEBRA. 

2.  To  find  the  lowest  common  multiple  of  polynomials, 
niustration.— Find  the  L.  C.  M.  of  a^  -  V^,  a"  -  ^>^  and 

Form. 

a^-i^  =(a-^b)(a-b) 

a^ -\- at  =  a  {a -{- h) 
L.  C.  M.  =  a  {a-\-h)  {a  -  h)  {a^ -{.ah-{-¥) 

Solution  :  To  contain  a?  —  b^,  the  L.  C.  M.  must  contain  the  prime 
factors  a  +  b  and  a  —  b,  and  to  contain  a^  —  b^  it  must  contain  the 
additional  factor  a^  +  ab  +  b^,  and  to  contain  a^  +  ab  it  must  con- 
tain the  additional  factor  a.  Therefore,  the  L.  C.  M.  is  a{a  +  b) 
(a-b){a^  +  ab  +  b^). 

EXERCISE    SS. 

Find  the  L.  C.  M.  of : 

1.  {a  +  by  and  a^  -  W  3.  o?  -  /  and  ar*  -  ^ 

2.  {a  —  by  and  a*  —  5*  4.  x^  —  y^  and  x^  —  y^ 

5.  x-\-y,  7?  —  y^,  and  o? -^-^xy  -\-y^ 

6.  x  —  y,  x^  —  y^,  and  a?  —  2xy -\-y^ 

7.  x^  —  y^,  x^  —  y^,  and  x^  —  2xy-{-y^ 

8.  {x -\-a){x-{-  b),  {x-{-a){x-\-  c),  and  (x  -\-b){x-\-  c) 

9.  a  (a  —  b),  b{b  —  a),  and  —  c(a  —  b) 

10.  {a  —  b){b  —  c),  {b  —  a){b  —  c)y  and  {b  —  a){c  —  b) 

11.  {x-{-yyy  ^'^  +  ^^  and  x^  —  y^ 

12.  a;2+5a:+6,  a;2  — 2a;  — 8,  and  a;2  — a;-12    • 

13.  a^  +  3a;-4,  x^-Qx-\-b,  and  a:^  _  ^  __  20 

14.  «  m  4-  ^  ^^  +  ^  ^^  +  ^  ^  and  ap-\-aq-\-bp-\-bq 

15.  ax  —  bx  —  ay-\-by  and  ax  —  ay -\-bx  —  by 

16.  «2  _  (J  ^  ^)2^  ^'  -  («  +  c)^  and  c2  -  («  +  ^)' 

17.  a3  +  3«2^'  +  3«Z»2_^J3  and  a^  _  ^  J2_^«2  j  _  53 


CANCELLATION.  93 

18.  x^-^t/y  a:*  +  ari/2_j-y4,  and  7?  —  y^ 

19.  2a:«  +  lla;  +  15,  ^7?-\-x-10,  and  x'-\-x-Q 

20.  6ar  +  13x  +  6,  6ar  — 5a;-6,  and  4:7?  —  ^ 

21.  ax^  —  ay^f  7^  —  xy^,  and  a.-^y  +  ^ 

22.  a;^  +  5  ^  +  6  ^^  7?y  —  7?  y  ^^xy,  and  7?y  —  ^y 

23.  2:^  +  2;^  +  ?/^,  7?  —  xy-\-y^,  and  re* +  ^2/^  +  ^ 

24.  2:^  +  /,  a:*  — ^^3/^  +  /,  and  a;^  +  / 


Cancellation. 

I.   Definitions  and   Principles. 

111.  Multiplying  a  quantity  by  a  factor  is  called  insert- 
ing a  factor. 

112.  Dividing  a  quantity  by  a  factor  is  called  eliminat- 
ing a  factor. 

113.  Crossing  out  a  quantity  and  writing  in  its  stead 
the  result  obtained  by  inserting  or  eliminating  a  factor  is 
Cancellation. 

114.  ah  Xac=^a^hc.  If  we  now  eliminate  a  from  a h 
and  insert  it  in  a  c,  we  have  h  X  oj^Cy  which  also  equals 
a*  h  c.     Therefore, 

Brin,  50. — Dividing  one  quantity  and  multiplying 
another  by  the  same  factor  does  not  alter  their  product. 

115.  ahcd  -^  ah  =  cd.  But  if  we  insert  l  in  ahcd 
we  have  a¥cd-i-ab,  which  equals  bed.  Also,  if  we 
eliminate  b  from  a b,  we  have  abcd-r-a,  which  equals 
bed.     Therefore, 

Prin,  SI. — Multiplying  the  dividend  or  dividing  the 
divisor  multiplies  the  quotient. 


94  ELEMENTARY  ALGEBRA. 

116.  ahcd-ir  ah  =  cd.  But  if  we  eliminate  c  from 
ahcdy  we  have  ahd-^  ah,  which  is  d.  Also,  if  we  insert 
c  m  ah,  we  have  ahcd-^  ahc,  which  is  d.     Therefore, 

Prin.  52. — Dividing  the  dividend  or  multiplying  the 
divisor  divides  the  quotient. 

117.  ahcd-T-  ah  =  cd.  If  we  now  insert  h  in  both 
abed  and  a h,  we  have  ah^ cd-^  aW,  which  is  c d.  Also, 
if  we  eliminate  h  from  both  ahcd  and  ah,  we  have 
acd-^  a,  which  is  c d.     Therefore, 

JPrin,  53, — Multiplying  or  dividing  hoth  dividend  and 
divisor  hy  the  same  quantity  does  not  alter  the  quotient. 

2.   Problem. 
To  multiply  or  divide  by  cancellation. 

niustratioiis.— 1.  Multiply  36  by  25. 

Solution :  Since  dividing  one  quan- 
tity and   multiplying  another  by  the  Form, 
same  factor  does  not  alter  their  product              9         ]^  QO 
[P.  50],  we  divide  36  by  4  and  multiply             *0  w  ^rt  __  qqq 
25  by  4,  and  obtain  9  x  100,  which  is  900. 

2.  Multiply  {a  +  hf  by  {a  -  h). 

Form, 

a-^h        a^-h^ 
(ar-\^f  X  («^--^)  =  a^-a¥-\-aH-h^ 

Solution :  Dividing  {a  +  hf  by  {a  +  b),  and  multiplying  {a  —  h)  by 
(a  +  I)  [P.  50],  we  have  {a  +  b)  x  (a" -  ¥\  which  \s  a^-ab'^  +  a'^b-  b\ 

3.  Divide  {a^  -f  ¥)  {a'  -  h^)  by  {a  +  hy. 

Form. 

(a^-\-h^){a^-h^)  _  (^H^)  (a^-ah  +  h^)  (^H-J)  (^  -  ^)  ^ 

a^-2a^h-^2ah^-h\ 

Solution :  Dividing  both  dividend  and  divisor  by  (a  +  b){a  +  b) 
[P.  53],  we  have  {a^  —  ab  +  b^) {a  —  b),  which  is  a^  —  2a^b +  2ab^ —  bK 


CANCELLATION.  95 

EXERCISE    86. 

Solve  by  cancellation  : 

1.  44  X  25  4.  42  X  16%  7.  26%  -r-  6% 

2.  36  X  15  5.  56  X  12%  8.  35%  -r-  7% 

3.  27  X  33%  6.  48  X  36  9.  144  -4-  36 

16  x  —  y^^' 

25X36  (,;  +  y)2(^_y). 

6%X7%  '^'  :^-y^ 

'^     2%,  "•       i^^T 

(a:»-.4)(a:^  +  6a:  +  8) 
(a^H-2a:-8) 

(g»+'7a;  +  12)(a:^  +  lla;  +  30) 
"•  (a;4-6)(a:2_^3^_^15J 

((x'-\-ac-^al-\-lc){W^lc-\-ld-\-cd) 
{al-\-ac-\-l^-\-ho){ah-\-ad'\'l)c-\-cd) 

(a  +  *  +  c)  (a  +  *  -  c) 

Find  the  value  of    the    following  expressions,   when 
a  =  10,  *  =  8,  c  =  6,  and  tZ  =  4. 

(a^-2gZ'  +  ^^)(a^  +  2g^  +  Z>g) 

a^-(bJrc)\a  +  h-\-c 
^'-  (^^{a-\-hf^  a-b-c 

^  {h-^cf-d^^  a-h-^c 

23  (^^-^^H^M:^)  ^      c^d 


(a-b){c-\-d)    ^a'-^ab^l^ 

\{a  +  bY^(c^dY]  \{a^bY-(c^dy\ 
(a-{-b-\-c-\-d){a-b-\-c-d) 


96  ELEMENTARY  ALGEBRA. 

Simultaneous  Numerical  Equations  of  Three 
Unknown  Quantities. 

Elimination   by  Addition  and   Subtraction. 

Direction  to  Pupil. — To  solve  three  equations  of  three  unknown 
quantities,  select  one  of  the  unknown  quantities  to  be  eliminated. 
Combine  any  two  of  the  equations  so  as  to  eliminate  this  quantity. 
Then  combine  either  one  of  these  two  with  the  third  in  like  manner. 
You  will  then  have  two  equations  having  only  two  unknown  quanti- 
ties, which  you  already  know  how  to  solve. 

Illustration. — Solve  : 


Solution 


6x-4:y-^2z=Q 

(A))- 

2x-{-Sy-4:Z  =  ll 

(B)y 

Sx  +  2y  +  6z  =  d7 

(C)) 

:  Multiply  (A)  by  2  and  bring  down 

(B), 

10x-8y  +  4z  =  12 

(1) 

2x  +  Sy-4z  =  n 

(B) 

Add  (1)  and  (B), 

12x-5y  =  2S 

(2) 

Multiply  (A)  by  5  and  (C)  by  2, 

25x-20y  +  10z  =  d0 

(3) 

Qx  +  4:y  +  10z  =  74: 

(4) 

Subtract  (4)  from  (3), 

19  a; -24  2/ =  -44 

(5) 

Multiply  (2)  by  24  and  (5)  by  5, 

288  a;  -  120  y  =  552 

(6) 

95a:-120y=-220 

(7) 

Subtract  (7)  from  (6), 

193  a;  =  772 

(8) 

a;  =  4 

*  Substitute  the  value  of  x  in  (2)  and 
y  =  5 
Substitute  the  values  of  x  and  y  in 

reduce, 

(B)  and  reduce, 

z  =  S 

Verification :  Put  4,  5,  and  3  for  x,  y,  and  z,  in  (A),  (B),  and  (C), 
20  -  20  +    6  =    6 ;  which  is  true. 
8  +  15-12  =  11; 
12  +  10  +  15  =  37; 

*  Substitute  means  put  in  place  of. 


CONCRETE  EXAMPLES,  97 

EXERCISE    87. 

Solve  : 

^-yj^z=    5V  3a;-4t/-f22=       4V 

x-\-y-z=    3)  5:r  +  3i/-7.  =  -16) 

2:^  +  3^-.  =  12)  8.5a;-62/  +  2.=    7| 

3^  +  3^  +  .  =  16)  2a:-3y  +  4.=  10) 

9.  2x-6y-3z  =  -14:) 
5x-7y  +  Gz=   4|  3^_4_5,  =  ^4 

2^  +  4^-5.=    6V  7^^6y+8.=     27 

a;  +  3i/  — 2;2  =  10  )  -^ 

*  10.  a:  +  ^  =    9  ) 
2:,_3y  +  4^  =  10)  x  +  z  =  10} 

3^  +  2;/-2.  =  17V  y  +  z  =  ll) 

x  +  5y-    z  =  22) 
^    ^  *  II.  x  +  y-z=    3) 

—  dx-\-2y-6z  =  —26\  x  —  y-\-z=    9V 

3x-2y-\-U=      21V  -  x-{-y +  z=ll) 

4:X  —  2ti—^z——2\)      ^    ^        .        ,  ^ 

-^  *12.  a;H-y  +  ;2;  =  6 

x  +  2y  — 3;z  =  — 1)  x-\-y^u=l 

4a;  — 4?/—    z=      8/-  a;+^  +  w=8 

3x-\-^y-\-2z  =  -b)  y-\-z-\-u=9 ) 


Concrete  Examples  involving  Simultaneous  Equa- 
tions of  Three  Unknown  Quantities. 

EXERCISE    88. 

1.  The  suDi  of  three  numbers  is  90 ;  twice  the  first, 
minus  three  times  the  second,  plus  four  times  the  third,  is 
200 ;  and  three  times  the  first,  plus  twice  the  second, 
minus  the  third,  is  10.     What  are  the  numbers  ? 

Suggestion. — Let  x  =  the  first,  y  =  the  second,  and  z  =  the  third. 

*  The  10th,  11th,  and  12th  are  most  readily  solved  by  comparing 
each  equation  with  the  sum  of  the  three. 


98  ELEMENTARY  ALGEBRA. 

2.  The  sum  of  three  numbers  is  90 ;  twice  the  first, 
plus  three  times  the  second,  is  30  less  than  four  times  the 
third ;  and  the  third  is  10  less  than  the  sum  of  the  other 
two.     Required  the  numbers. 

3.  A,  B,  and  0  together  have  $3500.  If  A  had  twice 
as  much,  B  three  times  as  much,  and  C  four  times  as 
much  as  now,  they  together  would  have  $9900 ;  and  twice 
C's  amount  exceeds  the  sum  of  A's  and  B's  amounts  by 
$400.     How  much  has  each  ? 

4.  A,  B,  and  0  together  have  500  acres  of  land ;  if  A 
buys  25  acres  from  each  of  the  others,  he  will  have  50 
acres  more  than  B  and  25  acres  less  than  C.  How  many 
acres  has  each  ? 

5.  The  cost  of  two  bushels  of  corn,  three  bushels  of 
oats,  and  four  bushels  of  rye  is  $5. 60  ;  of  three  bushels  of 
corn,  two  bushels  of  oats,  and  one  bushel  of  rye,  $3. 40  ;  of 
four  bushels  of  corn,  one  bushel  of  oats,  and  five  bushels 
of  rye,  $6.80.  Required  the  price  per  bushel  of  each  kind 
of  grain. 

6.  If  a  horse  and  cow  together  are  worth  $160,  a  horse 
and  sheep  $108,  and  a  cow  and  sheep  $68,  what  is  the 
value  of  each  ? 

7.  A  has  as  many  horses  as  cows  and  sheep  together ; 
twice  the  number  of  cows  is  12  less  than  the  number  of 
horses  and  sheep  together,  and  the  number  of  horses  and 
sheep  together  equals  four  times  the  number  of  cows.  How 
many  of  each  has  he  ? 

8.  The  sum  of  A's,  B's,  and  C's  ages  is  60  years ;  the 
sum  of  A's  and  C's  is  twice  B's  ;  and  C's  alone  equals  the 
sum  of  A's  and  B's.     Required  the  age  of  each. 

9.  A  man  has  two  horses  and  a  saddle,  together  worth 
$180.  The  first  horse  saddled  is  worth  twice  as  much  as 
the  second  horse  ;  the  second  horse  saddled  is  worth  $20 
less  than  the  first  horse.  What  is  the  value  of  each  horse 
and  the  saddle  ? 


CHAPTER   l(. 
ALGEBRAIC    FBACTIOJ^S. 


Preliminary  Definitions. 

118.  An  algebraic  fraction,  in  the  most  general  sense, 
is  an  expression  denoting  that  one  algebraic  quantity  is 
to  be  divided  by  another.  The  dividend  written  above  a 
horizontal  line  is  called  the  Numerator,  and  the  divisor 
written  below  the  line  is  called  the  Denominator. 

Illustration. — Thns,     ^,  read  a-\-h  divided  by  c  —  J, 

C        Uf 

in  which  a,  h,  c,  and  d  may  have  any  values,  positive  or 
negative,  integral  or  fractional,  is  an  algebraic  fraction. 

119.  The  numerator  and  denominator  are  called  the 
Term.s  of  the  fraction. 

120.  In  a  very  limited  sense,  in  which  the  terms  are 
restricted  to  arithmetical  integers,  an  algebraic  fraction 
may  be  defined  as  **a  number  of  equal  parts  of  a  unit." 

niustration. — Thus,  -r,  read  a  —  &th,  denotes  a  of  the 
h  equal  parts  of  a  unit. 

121.  Since  a  of  the  h  equal  parts  of  a  unit  is  equivalent 
to  one  of  the  h  equal  parts  of  a  units,  or  -^  of  1  =  ^  of  a, 
-T  in  the  more  restricted  sense  may  still  be  regarded  as  an 
expression  of  division. 


100  ELEMENTARY  ALGEBRA. 

122.  An  algebraic  fraction  is  usually  preceded  by  the 
positive  or  negative  sign  to  indicate  whether  it  is  to  be 
used  additively  or  suhtr actively, 

123.  The  value  of  a  fraction  is  the  result  obtained  by 
performing  all  the  operations  indicated. 

Illustration. — Thus,  the  value  of   —  -^  when  a  =  ~  6 

—  r> 
and  J  =  +  2  is  -—-=-(-  3)  =  +  3. 

-r  <> 

124.  The  apparent  sign  of  a  fraction  is  the  sign  pre- 
ceding it,  the  real  sign  the  sign  of  its  value. 

125.  An  Integral  Quantity  in  the  literal  notation  is  a 

quantity  that  is   not  fractional  in  form.     It  may  be  in- 

2 
tegral  or  fractional  in  value  ;  as,  a  =  8,  or  «  =  — . 

o 

126.  A  Mixed  Quantity  in  the  literal  notation  is  one 
that  is  partly  integral  and  partly  fractional  in  form ;  as, 

c 

127.  A  Proper  Fraction  in  the  literal  notation  is  one 
that  can  not  be  reduced  to  the  integral  or  the  mixed  form ; 

a  +  h 
as,  — ' — . 

c 

128.  An  Improper  Fraction  in  the  literal  notation  is 
one  that  can  be  reduced  to  either  the  integral  or  mixed 

form  ;  as,  — -^  =  a  —  o,  or  — ' —  =  a-\ — . 
a-\-h  a  a 

129.  A  Compound  Fraction  is  a  fractional  part  of  an 
integral  or  fractional  quantity  ;  as,  -^  of  c,  read  the  a  —  hth. 

ft  Q 

part  of  c ;  or  -^  of  -, ,  read  the  a  —  hih.  part  of  c  divided 

by^. 

130.  A  Complex  Fraction  is  a  fraction  one  or  both 
of  whose  terms  are  fractional  in  form. 


REDUCTION  OF  FRACTIONS.  101 

131.  The  inverse  of  a  fraction  is  the  fraction  resulting 
from  an  interchange  of  its  terms. 

Illustration. — Thus,  -  is  the  inverse  of  -7. 
a  0 

132.  The  reciprocal  of  a  quantity  is  unity  divided  by 
the  quantity. 


Reduction  of  Fractions. 

Definition. 

133.  Reduction  is  the  process  of  changing  the  form  of 
a  quantity  without  altering  its  value. 

Principles. 

134,  Since  multiplying  the  dividend  or  dividing  the 
divisor  multiplies  the  quotient  [P.  51],  it  follows  that, 

JPrin,  54. — Multiplying  the  numerator  or  dividing  the 
denominator  multiplies  the  value  of  a  fraction, 

a 

Bemark. — If  —  be  regarded  as  a  of  the  b  equal  parts  of  a  unit, 
b 
it  is  evident  that  multiplying  a  by  n  and  leaving  b  unchanged  will 
multiply  the  number  of  equal  parts  taken  by  n  without  altering  their 
size,  and  therefore  will  multiply  the  value  of  the  fraction  by  n. 

And,  dividing  6  by  w  and  leaving  a  unchanged  will  divide  the 
number  of  equal  parts  into  which  the  unit  is  divided  by  n;  and,  hence, 
make  each  part  n  times  as  great  without  changing  the  number  of  parts 
taken,  which  will  also  multiply  the  value  of  the  fraction  by  n. 

SIGHT      EXERCISE. 

Name  at  sight  the  products  in  the  following  examples  : 

1.  ^  X  c  4.  — ,  X  c  7.     2       ,8  X  (a  +  b) 

b  cd  a^  —  0^      ^  ' 

3.^x.       e.«^>(a-*)        ..,-Sx. 


102  ELEMENTARY  ALGEBRA, 

(a-\-'bY      ,        ,x  x  —  a       /     ,     X 

10-  5^.  X  («  -  i)  12.  ^-^^  X  («  +  «.) 

136.  Since  dividing  the  dividend  or  multiplying  the 
divisor  divides  the  quotient  [P.  52],  it  follows  that, 

JPrin,  55, — Dividing  the  numerator  or  multiplying  the 
denominator  divides  the  value  of  a  fraction, 

Remark. — If  —  be  regarded  as  a  of  the  b  equal  parts  of  a  unit, 
h 
dividing  ahj  n  and  leaving  h  unchanged  divides  the  number  of  equal 
parts  taken  by  n  without  changing  their  size,  which  evidently  divides 
the  value  of  the  fraction  by  n. 

And,  multiplying  6  by  n  and  leaving  a  unchanged  makes  the 
number  of  equal  parts  into  which  the  unit  is  divided  n  times  as  great, 

and,  therefore,  each  part  only  —  as  great,  without  altering  the  num- 
ber of  parts  taken,  which  divides  the  value  of  the  fraction  by  n. 

SIGHT      EXERCISE. 

Name  at  sight  the  quotients  in  the  following  examples  : 

a^  aH^         ^  m 

1.  T^  -r-  a  2. '-  ao  3.  — '-p 

¥  c  n     ^ 

4.  -  -^  V  10.  — -^-^  -^(xA-y) 

y  xy         ^     ^  ^' 

m^  -\-n^      ,  .  m^  —  n'^      ,  ^ 

7. , r-  (m  —  n)  13.  7 '-  {ni  —  n) 

m-\-n       ^  '  ao  ^ 

y    ^{x^-f)  14.  -—--^{x^Z) 


rg  +  2 
x-^Z  ' 


— ^  -^  («  4-  ^)  15.  -~  -^  {x  +  1) 


REDUCTION  OF  FRACTIONS.  103 

« 

136.  Multiplying   the  numerator  multiplies  the  value 

of  a  fraction  [P.  54]. 

Multiplying  the  denominator  divides  the  value  of  a 
fraction  [P.  55].     Therefore, 

Prin,  56, — Multiplying  hoth  terms  of  a  fraction  by  the 
same  quantity  does  not  alter  its  value. 

137.  Dividing  the  numerator  divides  the  value  of  a 
fraction  [P.  55]. 

Dividing  the  denominator  multiples  the  value   of  a 
fraction  [P.  54].     Therefore, 

JPrin,  57. — Dividing  both  terms  of  a  fraction  by  the 
same  quantity  does  not  alter  its  value. 

SJGHT      EXERCISE. 


Tell  at 
and  why  : 

;  sight  which  of  the 

following  equations  are 

true. 

-1= 

a^ 
ah 

«  «5 

2.  —  = 

ac 

5 
c 

3    ^^- 

a 
"h 

X 

4.  -  = 

y 

x(x^y) 

y{^-\-y) 

7. 

m 
n 

mn-\-m 
n^  -\-n 

X  —  a  _  a^  —  (^  c?  — '3?  _    a?'  —  7^ 

®'  x^a  ~  {x  +  af  ^'  oH^  ""  («M^ 

x-\-y  ^  rs  p'TS 
Complete  at  sight  the  following  equations  : 
10.^(^  +  ^U-  13.  ^'^*'- '- 


a{(i-\-x)      a  a^-h^      a"" -\- ah -\-l^ 

^^    ^^-y^  _y^-t  *       ^^  {m-\-n)a  ^m^n 
'  7?  —  y^           *  a{a  —  b) 

m      m^n^  r^s^a^ 

12.        = 15.  — ^  =  —5 

n  •  rs^q^      8^ 


104  ELEMENTARY  ALGEBRA. 

138.  Multiplying  both  terms  of  a  fraction  by  —  1  will 
change  the  signs  of  both  terms.  And  multiplying  both 
terms  of  a  fraction  by  the  same  quantity  does  not  alter  its 
value  [P.  56].     Therefore, 

Prin.  58. — Changing  the  signs  of  both  terms  of  a  frac- 
tion does  not  alter  its  value. 

a 


139.  Let  +  -^  =  +  ^ 
-a  a 

-  a 


then  -y-  or  —7  =  —  q  {F,  54] 

-,        —  a  a  /       x        .  .a 

and  ---^or  -  ~- =  ^  {- q)  =  +  q  =  J^ 


Therefore, 

Prin,  59. — Changing  the  apparent  sign  and  the  sign 
of  either  term  of  a  fraction  does  not  cha^ige  the  value  of  the 
fraction. 

Remarks. — 1.  Changing  the  sign  of  one  factor  of  either  term  of  a 
fraction  changes  the  sign  of  that  term.    Why  ? 

2.  Changing  the  sign  of  every  term  of  either  numerator  or  denom- 
inator changes  the  sign  of  that  term  of  the  fraction.    Why  ? 

SJGHT      EXERCISE. 

Change  at  sight  the  following  fractions  to  equivalent 
ones  having  apparently  positive  terms  : 

—  a                         a  —  h                                X  —  y 
1.  — ^  6.  11. ^ 


—  0  —  c  —z 

—  a  _         —a?  —x  —  y  —  i 

2. ^—  7.  —  — ^  12. ^ 

o  y^  xyz 

c  '         y{a  —  b)  '       —  (m^n) 

4    ^  9    -(a'  +  b^)  P  +  q 

-b  c^  +  d^  -ip^  +  q') 

—  a-\-b  xA-y  —  7n  —  n 
5.   ~-^             10.  =J-^-  15. 

—  c-i-d  —m  —  n  —x  —  y 


REDUCTION  OF  FRACTIONS.  105 

Change  at  sight  the  following  fractions  to  equivalent 
ones  having  positive  apparent  signs  : 

a(x-\-y) 


16. 

a  —  x 

a-\-x 

17. 

d  —  c 

io 

(a-h){c- 

-d) 

20. 


y  -X 
^^  +  / 

(m  -\-n)(m  —  n) 


{a^J)){c-^d)  ^^'  m^-\-n^ 


Problem  1.    To  reduce  a  fraction  to  its  lowest  terms. 

140.  A  fraction  is  in  its  lowest  terms  when  its  terms 

are  prime  to  each  other. 

c?  —  b^ 
Illustration.— Reduce    ..  ,   ,o  to  its  lowest  terms. 
fl^  +  ^"* 

Form. 

a^-b^  _        {^^^)  {a  -  b)         ^        a-h 

a^  +  b^"  (cC^^)(a^-ab^¥)  "  a^-ab-\-b^ 

Solution:  Since  dividing  both  t^rms  of  a  fraction  by  the  same 
quantity  does  not  alter  its  value  [P.  57],  we  divide  both  terms  by  their 
H.  C.  D.  (a  +  h).  The  resulting  fraction  is  in  its  lowest  terms,  since 
the  terms  are  prime  to  each  other.    Therefore, 

IRvle* — Divide  loth  terms  by  their  highest  common  di- 
visor. 

EXERCISE    89. 

Reduce  to  lowest  terms  : 

4:0,^1^  %x''y^z^  a  +  h 

^'  6aH  ^'  lOx'y  z^  '^'  a'-W 

^aH'(^  Hx±yY  {a  +  xf 

12aHV  ^{x^-yf  a^-^ 

Ibaf'y^^  ^'  aH{a?-y^)  (x-^%f 

^°-  {x-\-%yy  "•  (2a:-3y)« 


106  ELEMENTARY  ALGEBRA. 

'  2;*  +  2a;2«/2  +  /  ^"-    a;2  +  a;-20 

13  ^  +  y^  _-    0:^-12 a;  +  35 

4:3^- 26  y^  Sa^-27P 

^^  {2x  +  5yf  ^^'  (2a-3*f 

'  a?-{-xy-\-xz  {f-a^f 


{a-\-Vf-(?  a;«  +  ^/« 

ac-\-hc  —  G^  '  x^  —  y^ 


a^^  +  5^  +  6  ^j-2^V±/ 

ir2+7:i;  +  12  "       a;8-/ 

•         {a  -i){a  +  bf 

Problem  2.    To  reduce  a  mixed  quantity  to  an  improper 
fraction. 
lUustration. — 

^2  _|_  ^ 

Reduce  a-^-x to  an  improper  fraction. 


a-\-x  — 


a  —  X 

Form  and  Solution, 

a^-ira?      a  +  x      a^-\-x^      a^  -  x^      a^-\-a^ 


a  —  X  1  a  —  X        a  —  X        a  —  x 

[P.  56]  =  -A_  (^F^^  _  ^f+^)  [P.  47]  =  _!_(_  3^) 

-^^  ^^[P.59]. 


a  —  x  a  — a; 


EXERCISE    60. 

Reduce  to  improper  fractions  : 

,  a                    ,  x-\-a  ,       ,      a^ 

1.  a  +  —  3.  ^H ' — •  5.  a-\-x-\- 


X  '                   X  '11    ^_|_^ 

c  mx4-n  ^    ,  x  4- a 

2.  c 4.  m ' —         6.  1  H ' — 

y  X  X  —  a 


REDUCTIOIi  OF  FRACTIONS.  107 

.t2  „      ,   ^      3a-5 

7.  a  —  x i —  10.  2  a  +  7  —  -. r-x 

fl^  —  rr^ 


8.  a;  +  « — ^^  ^  11.  a^  +  a;?/4-y^-} 

12.  a;  +  7 


x  —  y 
3a;  +  5 


Problem  3.    To  reduce  improper  fractions  to  integral  or 
mixed  quantities, 
ninstrations. — 

1.  Reduce 7^     to  a  mixed  quantity. 


^-^+i_/„, 


Forait 


(a;2  -  a;  +  1)  ~  (2:  +  1)  =  a:  -  2  + 


Solution  :  Since  a  fraction  indicates  division,  and  the  numerator  is 
partly  divisible  by  the  denominator,  we  perform  the  division  and 
obtain  a  quotient  of  a;  —  2  and  a  remainder  of  3.  As  3  is  not  divis- 
ible by  a;  +  1,  we  simply  indicate  the  division  and  add  the  result  to 

Q 

x  —  %  which  produces  the  mixed  quantity  a;  —  2  + ~. 

^  A-x 4 

2.  Reduce  — — — z —  to  a  mixed  quantity. 

X  —  i 

Method. 

^'^^~'^  =  {x^-^x-^)---{x-l)  =  x-lr2-{-  ~^ 


x  —  1  ^'  ^  '  ^         '  x  —  1 

x  +  2-^[P.  59]. 

EXERCISE    61. 

Reduce  to  whole  or  mixed  quantities  : 

^   ac  +  b  ^     7?  ^   Q^-Yxy  +  y* 

c  x  —  1  '        x-\-y 

^  ax-a  ^-\-y^  Zx^-\-%x-Z 

2.    6.    ; 8.    -. — :; ■ 

X  ^-\-y  ^  + 1 

3.  ^^  +  a^  +  l  g   a^-y""  ^   a:*  +  ar/4-y* 

X  '    x-\-y  '    a^-\-xy-\-y' 


108                       ELEMENTARY  ALGEBRA. 
10.  '-^-  11. 12. 


x  —  y  %x  —  l  5a;— -6 

13.  — — ^ ^^—-  14.  ^ — ~- 

2x  2  a;  +  1 

Problem  4.    To  reduce  fractions  to  similar  forms. 
I.   Definitions  and   Principles. 

141.  Fractions  having  a  common  denominator  are 
similar, 

142.  Dissimilar  fractions  in  their  lowest  terms  must 
be  reduced  to  higher  terms  to  have  a  common  denomina- 
tor. This  is  done  by  mutiplying  both  terms  by  the  same 
quantity  [P.  56].  Therefore,  the  common  denominator 
must  contain  each  of  the  given  denominators.     Hence, 

JPrin,  60, — Any  common  multiple  of  the  denominators 
of  two  or  more  fractions  is  a  common  denominator  of  the 
fractions. 

Frin,  61. — The  lowest  common  multiple  of  the  denomi- 
nators of  two  or  more  fractions  in  their  lowest  terms  is 
the  lowest  common  denominator. 

Note. — L.  C.  D.  stands  for  lowest  common  denominator. 

2.  Examples. 
niustration. — 

Reduce  ^- ,  — ,  and  — r  to  similar  fractions. 
oc    ac  ao 

Solution :  The  L.  C.  M.  of  the  denominators  is  a  be,  which  is, 
therefore,  the  L.  C.  D.  [P.  60] : 

a  _  a  X  a    rp  f---.  _   a* 
h  c~  b  c  y.  a      '     ^~  abc 

ac      ac  X  b  ^         -'      abc 


c         c  X  c    ^-^  ^^,        c 


ab      ab  X  c  *-         -'      abc 
Note. — To  determine  the  factor  to  be  inserted  in  both  terms  of 
any  fraction,  divide  the  L.  C.  D.  by  the  denominator  of  that  fraction. 


REDUCTION  OF  FRACTIONS.  109 

EXERCISE    62. 

Reduce  to  similar  fractions  having  the  L.  C.  D. : 

X       y  ^    z  ^   ax    hx         ,  ex 

1.  —,.  -^,  and  T-  3.  7—,  — ,  and  -^ 
ah    ac            he  hy    ay^  ah 

a-\-h    a—  h        J    h  a  h  j<^ 

2.  — - — ,  ,  and  —        4.  — r^,  7,  and  -o — to 

xy  '     xz   '  yz  a-\-h'  a  —  h'  a^  —  W 

a  —  x    a  -\-  X            a^  —  TT 
a  h  ^  c 

3  ,  5 

1  1 

and 


^'  {x^af-W {x^hf-a^ 

2  3  4 

^*  (a:-l)(2:-2)'  {x-2){x^dy^^^  {x-l)(x-3) 

X  2x  —  l         -  2a;  +  l 

10.  7—2 7,  n rT»  ^"^  ^i 1* 

4:X^  —  1    2ic  +  l  2a;  —  1 

a  h ,  c 

"•  (a-c)(h-cy  (a-c)(c-hy  ^"""^  (c  -  a)  (c  -  h) 
Solation : 

^a-c){e-b)  ^  (a-c){b-c)  ^^'  ^^^  ^  ~  {a-c){b-c)  ^^'  ^^^ 
c c c 

(c  —  a){c  —  b)  ~  —  {a  —  c)  x  —{b  —  c)  ~  {a  —  c){b  —  c) 
a a 

{a  —  c){b  —  c)  ^  (a  —  c){b  —  c) 

ah  c  ^34  5 

12. ,    -,    ;; T  13. 


l-.x'x-Vl-x'       *"•  2-a;'  a;-2'  (x-2)« 
1  2,3 

1^    7 TV7Z \»  7 SVTo \'  ^^^ 


(X  -l)(^^xy  {x-  2) (3  -  xy  ""^  (1  -x)(x-  3) 
15.  7 w r,  7 r-7T r,  and 


{a-x)(x-cy  (x-a){h-xy  (c-x)(x-h) 

6 


110  ELEMENTARY  ALGEBRA. 

Addition  and  Subtraction  of  Fractions. 

1.  Principles. 

^^^•1  +  2+2  =  1^''  +  ^  +  '^  [P.  47]-^Li.. 
Therefore, 

JPrin,  62, — The  sum  of  two  or  more  similar  fractions 
equals  the  sum  of  their  numerators  divided  hy  their  com- 
mon denominator, 

144.  ^  _  1  =  1  (^  _  J)  [P.  47]  ==  ^Jzi.     Therefore, 
c       c       c  ^  ^  ^         ^  c 

Prin,  63. — The  difference  of  two  similar  fractions 
equals  the  difference  of  their  numerators  divided  hy  their 
common  denominator, 

2.  Problem. 

To  add  or  subtract  fractions. 

Dlustrations.— 1.  Find  the  sum  of  and  — ; — . 

a  —  x  a-{-x 

Solution  :  The  L.  C.  D.  =  a^  -  x^ 

a  +  X _{a  ■{■  x) (a  +  X)  _a;^  +  2 ax  ■\-  x^ 
a  —  x~  {a  —  x) {a  +  X)  ~        o*  —  a;* 
a  —  x      (a  —  x)  (a  —  x)      a^  —  2ax  +  x^ 


a  +  X      {a  +  x)(a  —  x)  a^  —  x^ 

a''  +  2ax  +  x^      a^-2ax  +  x^  _2a^  +  2x^ 

a  h         a^^-h^ 


[P.  62] 


2.  Find  the  value  of 


Solntion 


a  — I)      h  —  a      a^  —  W 

a(a  +  b)  a^  +  ab 


a-b      {a  —  b){a  +  h)       a^  -  b'^ 

h     _  b  b{a  +  b)      _ab  +  b* 


b  —  a  a  —  b      {a  —  b)(a  +  b)      a^  —  b* 

_aM^_      -  CT^  -  6» 
a2  _  53  -  +    a2  -  62 

a^  +  ab      ab  +  b^      -  a^  -  b^      '2ab 


+   -5 T^  + 


J2  -r  a«-62  ^     a^-b^    ~a^-b^ 


ADDITION  AND  SUBTRACTION  OF  FRACTIONS.  \\\ 


Note.— Sometimes  it  is  better  not  to  combine  all  the  fractions  at 

one  time. 

3.  Find  the  value  of  ; 

l+a;"^l 

1 

—  X 

1 

l+a;^- 

8oliiUon:^^^  +  ^_^  = 

\-x       1 

+  x 

2 

2              1 

2  4-2a;« 

1- 

X*      l  +  3a;« 

1-x^      1  +  x^ 

~  l-o;* 

1- 

a;*"  1-a:* 

4.  Find  the  sum  of  2 

Method. 

2«- 

—  and  3  a 

;  +  2J 

3«-5 
c 

2a     36  +  ^''" 

l^  =  2a- 

36  + 

2a-6 
c 

8a  +  2&     ^"^ 
c 

l^  =  3a  + 

26  + 

-3a  +  6 
c 

Sum 

=  5a- 

b- 

EXERCISE    63. 

Find  the  value  of  : 


1  +  a      1  —  a 
a  a 


a-\-x     a  —  x 

ah  ,  ac  ,  he 
c    '    h    ^    a 

1,1        1 
ao     ac      be 

^a±z_a±y  ^^ 

X         y  p-1    q—p 

-(¥+'4*)-(t-'-I*) 


x'^f 

^^-f 

m 

n 

m  —  n 

n  —  m 

P      1 

p> 

p-q^ 

f-f 

"      1 

a             c? 

a-x  1 

a-\-x      x^~ 

a« 

o  +  i 

a-h 

112  ELEMENTARY  ALGEBRA. 

ic    /^  +  ^  ,   c^d\       (a-h  ,   c-d\ 

\x      y  ^  zj^\x       y       z) 

'•  V3cB"^4y      3«y"'"l^3a:      3y+4«^ 
■  \3a;      5y'^6z)       \5x^  S.y      4.z) 


a  +  a;  a  — a;  {a  +  xf  '  (a  —  :z;)^ 

20.  ^  I       ^y  I         ^-y 

X  xy              7? 

21.  '^ 


22.  ^^ + I + I— 

x^  —  dx-\-^^x^-4.x-\-Z^x''  —  bx-\-Q 

2 4  2 


24. 


a?-^x-\-\%      a?-^x-\-^^  T^'-hx-^^ 
-1 + 1 


2  3  2a;  — 3 

25. 


26. 
27. 


ic      2a;-l      4a;2-l 

3  +  2a;      2-3a:      16:r  — a^ 
2  -^  aj         2  +  a;  "^    a;^  _  4 

3 y  4 -20a; 

l-2a;      l  +  2a;       4a;2-l 


MULTIPLICATION  AND  DIVISION  OF  FRACTIONS.  113 

Multiplication  and  Division  of  Fractions. 

Problems. 

1.  To  multiply  or  divide  a  fraction. 

niustrations. — 1.  Multiply        .     by  a  —  h. 

»  ^  .-        a  +  h      .        ,,      (a  +  h){a-h)  a^-b^ 

Solution:  -^  x  (a-h)  = ^ [P.  54]  =  -^^ 

2.  Multiply  ^2^_^  by  a  +  5. 

solution:  ^^  x  (a  +  5)  =  ^-^_^,g^^^  [P.  54]  =  ^ 

3.  Multiply  |±|  by  a^  -  h\ 

Solution :  ^  X  (a*  -  &»)  =  ^^  "^  ^^^'^  [P-  57]  =  (a  +  6)« 

4.  Divide  7 —  by  a-\-'b, 

ah       ^       ' 

Solution :  j—  -s-  (a  +  6)  =  ^ '-^ [P.  55]  =  — 1— 

ab        ^         '  ao  ^        ^        ao 

5.  Divide  — tt:  by  a  —  h, 

6.  Divide  ^'7^^  by  a^-^. 

Solution:  t^— r h- (a«  —  6«)  =  tts— — rrr-i — j:5^  = 

o(<r^^) CT 

ft(a  +  ft)(S^^(a  +  6)  ~  6(a  +  ft)« 

7-  J^2-(^  +  ^)X(c  +  ^)  = 

a  —  h^,     I    7\      « —  ^ 


114  ELEMENTARY  ALGEBRA, 

EXERCISE    64. 

Multiply  :  Divide  : 

1.  —  by  «^  9.  by  ac 

2.  -y^-g  by  de^  10.  —  by  cc? 


a^<^ 


3.  To-^  by  c^S^  11.  -^^  by  Sa;^^^ 

^  ^7^  by  ^y  12.  -^by:r  +  y 

6-    3  .  ra  by  a  +  ^  14.  -^ by  4:(m^4-n^) 

''■  «.-^:T^  by  («  + J)^         15.         ^_^l       by  x  +  5 

17.  Multiply  g;gg;g  by  (.  +  5)^' 

18.  Multiply  g-lj^  +  gl-  hjx-a-^-b 

19.  Divide  ^^'^(^  +  ^)  by  6ci;(a;-;2) 

20.  Divide  5 by  a-{-o-{-c 

Simplify : 

22- ^^-(«+y)x(a:-y) 


MULTIPLICATION  AND  DIVISION  OF  FRACTIONS.  115 

2.   To  multiply  or  divide  by  a  fraction. 

I.  Definitions  and   Principles. 
145.  To  multiply  or  divide  by  a  fraction  is  to  multiply 
or  divide  by  the  quotient  of  two  quantities. 

Illustrations. — 1.  Multiply  j  by  ^. 

Solution  :  Since  multiplying  one  quantity  and  dividing  another  by 
the  same  factor  does  not  alter  their  product  [P.  50], 

[P.  54].    Therefore, 

Prin,  64* — The  product  of  two  fractions  equals  the 
product  of  their  numerators  divided  hy  the  product  of  their 
denominators, 

SIGHT      EXERCISE. 

Name  at  sight  the  product  in  the  following  examples  : 
X      m  T^      7?  X      m^ 

y      n  y       T  y^      n^ 


4. 


a_±x      a-x  /     mn\       (     pq\ 


6. 


£+2      £+1  a(m-n)      (m-ny 

'^'  x-^S^x-S  *  (m  +  w)  ^  (m  +  ny 

2.  Multiply  |-'x^'. 

Solution  :  Since  multiplying  one  quantity  and  dividing  another  by 

the  same  factor  does  not  alter  their  product  [P.  50],  we  multiply  the 

first  fraction  by  ft'  by  dividing  its  denominator,  and  divide  the  second 

a^      1 
fraction  by  a'  by  dividing  its  numerator,  and  have  remaining  jj  x  ^» 

a' 

which  equals  rs-s .    Therefore, 

0  c 

Prin,  65. — Canceling  a  factor  common  to  the  numerator 

of  one  fraction  and  the  denominator  of  another  does  not 

alter  the  product  of  the  fractions. 


116  ELEMENTARY  ALGEBRA. 

SIGHT     EXERCISE. 

Multiply  at  sight  the  following  fractions  : 

^  a-l  ^  a^-y^  ^    (^ -\- yf      x-y 

'x-\-y      a^  —  W  '  {x  —  yf      x-\-y 

a{x-\-y)      b{x^-y^)  _^v^ 

^'  h{x-y)^         a  ^'       y^^a^ 

{a^^y      (a-Zf  y-x      z-x 

^-  {a-Zf^  {a-\-^f  ^'  x-z  ^  x-y 

3.  Divide  ^  -^  -^  • 
h       d 

Solution :  Since  multiplying  both  dividend  and  divisor  by  the  same 
factor  does  not  alter  the  quotient  [P.  53], 

[P.  14].    Therefore, 

JPrin.  66, — The  quotient  of  two  fractions  equals  the 
dividend  multiplied  hy  the  inverse  of  the  divisor, 

SIGHT      EXERCI  SE. 

Name  at  sight  the  quotient  of  the  following  fractions  ; 

X      m  a?      y  ah       c 

1.  --T-—  2.  ^-v-^  3.  — ^-^^ 

y      n  y^      X  cd      d 

a^  —  x^  ^  a  —  x  Inn?      v\  .  "^P 

X       '      ^  '  \n^      'qj  ~  'nq 

mnp  ,    qrs  a{x  —  y)  ,       1 


6.  ^  -^  ^ 10 


qrs       m  np  1  x  —  y 

%       xy  a^  —  Wa  —  l 

xy       %  on?  —  y^  '  X  —  y 


MULTIPLICATION  AND  DIVISION  OF  FRACTIONS.  117 
4.  Divide  by  — 5—. 

X  XT 

Solution:  Since  dividing  both. dividend  and  divisor  by  the  same 
quantity  does  not  alter  the  value  of  the  quotient,  we  diyide  both 
fractions  by  a  —  h  by  dividing  their  numerators  by  a  —  h,  and  obtain 
0  +  6      1  Since  multiplying  both  dividend  and  divisor  by  the 

X         X* '     same  quantity  does  not  alter  the  value  of  the  quotient, 
we  multiply  both  fractions  by  x  by  dividing  both  denominators  by  x, 

and  obtain  — z 1 — ,  which  equals  {a  +  b)x  [P.  6C].    Therefore, 

Prin,  67 > — Canceling  a  factor  common  to  the  numer- 
ators or  the  denominators  of  two  fractions  does  not  alter 
their  quotient, 

SIGHT      EXERCI  SE. 

Name  at  sight  the  quotients  of  the  following  fractions  : 


a^      a 

x^ 

7? 

■  x^y^  '  xy 

fl2  _  ^      a  -  a: 

T^ia-^xY  .  (a  +  xY 
^'  s'ia-xY  '  {a-xY 

xy             X 

(x-^-'^Y      a;H-2 

m^  -\-n^      m-\-n 
mr  —  n^      m  —  n 

a;  +  3     *  x-\-Z 

a{p  +  q)  ,  p-^q 

(x  +  %)(x-Z)  .x-Z 

b(p  —  q)      p  —  q  {x—2){x-{-d)'X'-'2 


WRITTEN      EXAMPLES. 

Ulnstrations.— 1.  Multiply  -TZTRt  ^^  ~T~~* 

1 

Solution :  -, — p,  x  — —  =  ^^    . ,  x  ^!!^  [P.  65]  =  —. ^r . 

a  —  b 

2.  Divide  7—  by  ^ . 

cd        -^        de 

a-\-  b 
solution:  -^ +  ^_  =  _^^  J^[P.  67]  = 

i±*xl[P.66]=^<!i±l>, 


118  ELEMENTARY  ALGEBRA, 


3.  DiYide  1+-  by  1-- 

y   ^       y 


Solution 


1  +  —  (l  +  — )  X  y 

^  V      y)  '  \      y)     i_£  (i-^)x2/    2^-^' 

Or,  y  \      y)     ^ 

\      y)     \      y)       y    '     y  y      y  —  ^    y—x' 

EXERCISE    68. 

Find  the  yalue  of  : 

ax      ex  a-\-x      a^  —  x^ 


hy      dy  cd  de 

2^3 


2.  — ^-^  X   ^  «*  


«*a;* 


xy^z^^  a^h^i?  a^^o?  '  a^  ^  x^ 

ax  hx  ^^   a^-\-x^  ^  a^  —  x^ 

^'    '^  a-\-x  '    a  +  y    '  a^'-y^ 


a 


a" 


ax 


X 


11. 


X  p-q    p-\-q 


^V^V^  12  (^  +  ^)'x^'-y' 

by^dx^ad  ^-y       aJ'  +  ab 

Ji/^  *  dy^ 


k-M^-S} 


16. 


a;2  +  7a;  +  10  *  a:2-a:-30 


ac  —  ad  —  bc-\-bd      c-\-d 


18. 


a:3-f27f/^   a;^  -  3a;y  +  9.y^ 


COMPLEX  FRACTIONS,  119 

Complex   Fractions. 
To  reduce  a  complex  fraction  to  a  simple  one. 
a 
Illustration. — 1.  Reduce  —  to  a  simple  fraction. 


a       a      .  ,  a 

«...         ^       *■  ad      ^      h       a       c       a       d      ad 

Solution :  —  = =  -r— ;    Or,  -.  =  ^^  —  =  — x—  =  -j— . 

c       c       ^  ,       be  c       0       d       b       c       be 

d-d.  d 

2.   Reduce to  a  simple  fraction. 

3^  —  — 

Solation : 

ic' _  \         a:' /      _x^  —  X 

^      V         ^V  a;(a:«  +  l) 

Or,  a;*  +  a:«  +  1  "**** 

»_1      ^-^  a: 

^ £__fL-?llll     ^•g^. a;(a:«+l)(a;4-l)(a;~l)  _ 

1  ~a;«-l~     ^«     ^a^-l~(a;+l)(a;«-a;+l)(a;-l)(a;«+a;+l)~ 

EXERCISE    66. 


a;(a;»  +  l)((^H4)(^^) 


Simplify  : 

a 

^  +  1 

X 

''1 

y 

x+i 

ax 

hy 
*2.  -i 

xy 

ab 

a 

c 

5. 

a;  +  2 

6. 

a:-3 

a:-2 

.T  +  3 


a-f 

■x 

7. 

a: 

a  — 

a 

8. 

1 

"l 

X 

1  + 

120  ELEMENTARY  ALGEBRA. 

1 


X  1 

9.  11.  a:  4 


x-\--  x-\-- 

X  ^    X 

,  ^  ,       1  .2a 


x—%  ' «— 3 

10.  12.   

„   ,       1  2a 

x-{-Q  a  — 3 


Involution  of  Fractions. 

146.  Involution  of  fractions  is  the  process  of  raising  a 
fraction  to  any  power. 

Problem. 
To  raise  a  fraction  to  any  power. 

Illustration. — 1.  Raise  -r  to  the  fifth  power. 

Solution  '(j)=j^j^jXj^j  =  Y.-    Therefore, 

Prin,  68. — Raising  both  terms  of  a  fraction  to  any 
power  raises  the  fraction  to  that  power. 

SIGHT     EXERCI  SE. 

Name  at  sight  the  indicated  powers  of  the  following 
fractions  : 

'■©■      -(-f;)"     HffefF 

\by/  \      Pq)  \     ^^  P ) 

(     ^2X4  (a-bV  I     r^s^'V 

*■  \3¥)  ^'  \x^y)  ^^-  \     mH^j 


INVOLUTION  OF  FRACTIONS,  121 

WRITTEN      EXERCISES. 
EXERCISE    67. 

Find  the  value  of  : 

■•(14)"       A'§m 

Miscellaneous  Examples. 


EXERCISE    68. 
/p2  a;  30 

1.  Reduce     «         — — r^r  to  its  lowest  terms. 

7^  —  1^ 

2.  Reduce  -» — ^  to  its  lowest  terms. 

7^  —  y^ 

3.  Reduce  r^^  ^r-^r — rTTTZJ  ^  i^  lowest  terms. 

ac-\-Sad-\-5bc-{- 16  bd 

4.  Reduce  1  —  \o.     2  ^^  ^^  improper  fraction. 

(x  —  yV 
6.  Reduce  2x  —  y-{- ^^  to  an  improper  fraction. 

1  -_  2ar 

6.  Reduce  — — ; to  a  mixed  quantity. 

1-^x 


122  ELEMENTARY  ALGEBRA. 


7.  Simplify       3  ^  4-20« 


8.  Simplify 


l-2a      l  +  2a       4a2_i 

fl^^  +  ^'  _  _«" ^__ 

ah  al^W~  a^^ah 

2 


9.  Simplify  ^±i.:+ii^,i^i^ 

10.  Simplify  -^  X  L_^ 

— «{'+(^)M-(sl)l 

.,.M.„iplj|  +  £b,J  +  | 

^  Operation. 


a 

c 

J 

+ 

d 

c 

d 

— 

4- 

h 

a 

c 

c^ 

J 

+ 

ad 

c. 

+ 

b^ 

+ 

h 

2e 

c^ 

ad 

T 

+ 

^ 

+ 

-¥- 

14.  Multiply  I- I  by  ^  +  ^ 

15.  Multiply  ^  +  f  by  ^^  +  g 

16.  Square  -  + 1 ;  also  -  -  ^    [See  P.  31,  32.1 

a   ^    h^  y      a     ^  -■ 

17.  Cube  1  +  - ;  also  --b    [See  P.  33,  34.1 


18. 


Find  the  value  of  (^  +  ^ W^  -  ^^     [See  P.  39.] 


19. 


20. 


21. 


MISCELLANEOUS  EXAMPLES.  123 

Find  the  value  of  (^  +  ^)  [^  —  ^) 
Find  the  value  of  f  a;  +  1  +  -  j  ;   ix  —  l-A 
Find  the  value  of  (^±^  +  ^)  (^±^  _  ^) 


22.  Divide  -§  —  ^  by y    [See  P.  44.] 

23.  Divide  T5  +  -3  by  y  +  - 

24.  Divide  ^--,  by  ---;  also  by  -  +  - 
26.  Factor  a^  -  ^  ;  a^f--^;   h^~^^^ 

26.  Factor  1-^;  «*--,;  -,-^ 

27.  Factor  2;'  +  ^;  ^-^5   ^"7 

28.  Factor  ^  +  '^;  — «  -  ^  ;  l  —  lr-Ml 

y^  ^  x^^  m^      n®  S^  —  yJ 

29.  Divide  a:'^  +  -5  by  a:  +  - 

30.  Find  the  value  of  : 

a;  +  y  a;-  +  / 

^      X  —  y   '      __x^  —  y^ 


x-\-y  x^  +  y^ 

-,,,.,     a:      a   ,   y      h  .      x      a      y  ,   b 

31.  Multiply Kt by t  +  -  as  m 

^•^a      x^  b      y         a      X      b   ^  y 

multiplication  of  entire  polynomials. 

32.  Find  the  value  of  ^ when  z  = 


b  —  z  a-\-b 


CHAPTER    111. 

GEJ^ERAL     TBEATMEJ^T    OF    SIMPLE 
EQUATIONS, 


General   Definitions. 

147.  An  equation  in  which  the  known  quantities  are 
numerical  is  a  Numerical  equation  ;  as, 

dx-{'2  =  6x-4:. 

148.  An  equation  in  which  some  or  all  of  the  known 
quantities  are  literal  is  a  Literal  equation  ;  as, 

2ax  —  4:bx  =  c. 

149.  The  degree  of  a  term  of  an  equation  is  determined 
by  the  number  of  unknown  prime  factors  it  contains. 

Thus  :  In  ax^-\-bxy-\-cy^-\-dx-\-ei/ -\-f  =0,  a oi^, 
hxy,  and  cy^  are  of  the  second  degree;  dx  and  ey  oi 
the  first  degree  ;  and  /  of  no  degree. 

150.  A  term  in  an  equation  that  does  not  contain  an 
unknown  quantity  is  an  Absolute  term. 

151.  The  degree  of  an  equation  is  the  same  as  the  degree 
of  its  highest  term. 

152.  An  equation  of  the  first  degree  is  a  Simple  equa- 
tion ;  as,  2x  —  Sx-{-5x=zl2  ;  or,  ax -\- by  =  c, 

153.  An  equation  of  the  second  degree  is  a  Quadratic 
equation  ;  as,  a^-{-4cX  =  6  ;  or,  x^  -{- xy  =  12, 

154.  An  equation  of  the  third  degree  is  a  Cubic  equa- 
tion;  SLS,x^  +  dx^-{-2x  +  6  =  0',  or,  a^  +  x^y -\- x -{■  y  =  12. 


TRANSFORMATION  OF  EQUATIONS,  125 

Transformation  of  Equations. 

Definition  and    Principles. 

155.  The  process  of  changing  the  form  of  an  equation 
without  destroying  the  equality  of  its  members  is  trans- 
formation of  equations. 

166.  An  equation  may  be  transformed  : 
Prin,  69. — 1.  By  adding  the  same  or  equal  quantities 
to  both  members. 

2.  By  subtracting  the  same  or  equal  quantities  from 
both  members, 

3.  By  multiplying  both  members  by  the  same  or  equal 
quantities. 

4.  By  dividing  both  members  by  the  same  or  equal 
quantities, 

6,  By  raising  both  members  to  the  same  power, 
6,  By  taking  the  same  root  of  both  members. 

157.  If  we  take  the  equation 

ax  —  b  =  cx-\-d,  (1) 

and  add  b  to  both  members  [P.  69,  1],  we  obtain 

ax  =  cx-{-  d-\-b,         (2) 
K  we  now  subtract  c  x  from  both  members,  we  obtain 
ax  —  cx=.d-\-b.  (3) 

If  we  now  compare  (3)  with  (1),  we  observe  that : 

Prin,  70, — Any  term  of  an  equation  may  be  transposed 
from  one  member  to  the  other  if  its  sign  be  changed. 

158.  If  we  take  the  equation 

|  +  ¥  =  T-^'  (1) 

and  reduce  all  the  terms  to  a  common  denominator,  we 
,  ^  .  6  a;  ,  4  a:      15  a;      36  ,„, 


126  ELEMENTARY  ALGEBRA. 

If  we  now  multiply  both  members  by  18  [P.  69,  3],  we 
obtain  6a;  +  4a;  =  15ic  —  36,  an  equation  without  frac- 
tional terms.  But  18,  the  common  denominator  of  the 
fractional  terms,  is  a  common  multiple  of  the  denomina- 
tors of  the  fractions.     Therefore, 

Trin,  71* — An  equation  with  fractional  terms  may  be 
cleared  of  fractions  hy  multiplying  loth  members  by  a  com- 
mon multiple  of  the  denominators  of  the  fractions. 


Simple  Equations  of  One  Unknown  Quantity. 

I.   Solution  of  Numerical  Equations. 
Illustrations. — 

1.  Given  hx-\-^  =  Zx  —  h-^^x  to  find  the  value  of  x. 
Solution:  Src  +  7  =  3ic  — 5  +  6a;  (A) 
Transposing  7,  3a^,  and  6  a;  [P.  70], 

5a;-3a;-6a;=-5-7  (1) 

Uniting  terms,  —  4  a;  =  —  12  (2) 

Dividing  by  -  4,  a;  =  3  [P.  69,  4]. 

Proof :  Substituting  a;  =  3  in  equation  (A) 

15 +  7  =  9  —  5  +  18;  whence 
22  =  22. 

Q  /v  ^  T  ^  '7'  7 

2.  Given  ~ 4  +  -^^  =  —-  +  - to  find  the  value  of  x. 

<i  o  4         o 

a  ,   ,.  3a;       .      7a;      5a;       7  ,.. 

Solution:  __4  +  — =— +  -  (A) 

Clearing  of  fractions  by  multiplying  by  12  [P.  71] 

18a; -48  + 28a;  =  15a;  +  14  (1) 

Transposing  —  48  and  15  x  [P.  70] 

18a;  +  28a;  -  15  a;  =  14  +  48  (2) 

Uniting  terms,  31  a;  =  62. 

Dividing  by  31,  a;  =  2. 

Proof :  Substituting  a;  =  2  in  equation  (A), 

o      ^      14      5       7 

3-4  +  —  =  -  +  -;  whence 

11_11 
3  ~  3  ' 


SOLUTION  OF  NUMERICAL  EQUATIONS.        127 

3.  Solve  X g —  =  2 g-. 

Solution :  Giveu  x ^ —  =2 1^-  (A) 

Clearing  of  fractions  by  multiplying  by  12, 

12a;  -  2  (3a;  -  2)  =:  24  -  (a;  +  C)        (1) 
Expanding,  12  a;  -  6  a;  +  4  =  24  -  a;  -  6  (2) 

Transposing  and  uniting  terms,   7a;  =  14  (3) 

Dividing  by  7,  a;  =  2. 

Equation  (1)  may  be  omitted  if  the  following  principle  be  heeded : 

159.  If  a  fraction  is  preceded  hy  minus,  the  sign  of 
every  term  in  the  numerator  must  he  changed  when  its  de- 
nominator is  removed.    Why  ? 

EXERCISE    60. 

Solve  the  following  equations  : 

1.  5a:  +  3  =  7a;-3  2.  3a;  + 5a;  + 14  =  9:r-|- 10 

3.  x-\-7-Sx  =  5-(j{x-l) 

XX      X  __                         x-^h      x-e_x  ,   ^ 
^2  +  3-4-^  ^-  ~3 4--6  +  ^ 

X      X  ,    X              „^           ^         2.T  — 3      ^^      2x 
'^•4-6  +  r2  =  ^-^^       '-^^ 2-  =  ^^-T 

8.  3(x-\-5)-4:(x  +  Q)  =  x-U 

9.  x-l(3x  +  6)  =  l{6x-10) 

o  o 


10.  3a:  —  (2a:  —  a;  —  2)  =  3a;  —  9 

K  10 

11.  3a:-^a:+7  =  -^a:-2 

o  o 

15a;  — 6a:      ^      ^      18ar 

12.-^—8  =  7-^ 

5a:  — 6      7a:--3_2a:  +  8 
5      "^      10     ""      15 

1— a:      X  —  1  _x-{-l 
^^  "~3  6~""32^ 


128  ELEMENTARY  ALGEBRA. 

160.  It  is  sometimes  better  to  unite  some  terms  before 
clearing  of  fractions.     Thus, 

G.^en  -^ ^  =  -10 10 5 

Eeducing  to  common  denominators, 

2a;  +  12      2a;  — 5_8a;  +  14      7a;  +  4       6 
8  8       ~       10  lO  10 

Uniting  terms,  —  =  -^ 

Clearing  of  fractions,  8 :?;  +  32  =  170, 
whence  ^x  =  138 

and  ic  =  17V4 

15.  Solve  — ^ _  =  -X__|._ 

^    ^  ,       3a;-2   ,   18      5ir+3       1 

16.  Solve  -^+j9  =  -f^-j-, 

17.  Solve  — ^ 1-         '      —         '        ' 


18.  Solve 

19.  Solve 


5  '        8                5        '        8 

Z  —  x  ^-\-x_      5 

3  +  a;  Z  —  x~Q^—^ 

x-\-%  x  —  d          x  —  d 


x-\-3      x  —  2      x^  +  x  —  e 


oil  1  «+l 

20.  Solve  —     ^ 


2      x-^2      a^-4: 

21    SolTC^^  +  ^'/^      2rr-2_4a;-2 

21.  bolve         ^  7a:- 6-      10 

-^   a  1      6a;  +  7      a;  — 5  ,  2a;  — 3 

22.  Solve 


23.  Solve 

24.  Solve 


18 

x-6   '        6 

7a:-5 

a;— 2      5a;  +  6 

21 

3           a;-3 

3a;-2 

5 -9a;      a;  -  7 

3 

'        9       ~a;  +  7 

SOLUTION  OF  SIMPLE  LITERAL  EQUATIONS.    129 
2.   Solution  of  Simple  Literal  Equations. 

X         X         d 

Illustrations. — 1.  Solve  -  -I-  ^  =  -. 
a  ^   0       c 

Clearing  of  fractions, 

hcx  +  acx  =  a^h 

Factoring,  {he  +  ac)x  =  a^b 

Dividing  by  coeflBcient  of  a;,       a;  =  t — - — - 

2.  Solve  —!— +  — 4-  =  — ^. 
a  ah  0 

Clearing  of  fractions, 

ab  +  lx  +  b  +  x  =  ac  -^  ax 
Transposing,  bx  —  ax  +  x  =  ac  —  ah  —  h 

Factoring,  {b —  a  +  \)x  =  ac  —  ah —  b 

Dividing  by  the  coefficient  of  a;,  x  =  —r r— 

0  —  a  +  1 

EXERCISE    70. 


Solve 


1. 

X    ,    X 

2. 

ax  —  bx  =  a  —  b 

3. 

cx-\-dx=:c^  —  d^ 

4. 

m^x  —  n^x  =  m*  —  n* 

6. 

a-\-x      a  —  x      a 

b      ^       c     ~c 

6. 

a—x      b—x        c 
m            n         mn 

X  X  ^ 

a-\-b      a  —  b 

ab  —  ex  _  cd -\-ax 
d  a 

a  —  mx      c  —  nx 


12. 


1,1,1       1 

13.  -  +  -  +  -  =  - 
a      b       c      X 

a  ,   b      c      d 

14.  ^H 3  =  - 

b      c      d      X 


^    „         ax      "^ax  ,   _  a  b 

7.  3  a;  —  -^  =  —^-  +  2  16.  z r-  =  q 

2  3  l^bx      1  —  ax 

a  —  x      ^      .  b-\-x  ac      be  ,   , 

8.  X -— =  2a:  +  — ^  16.  T =  a-\-b 

5  '10  bx      ax 

mx  —  nx  _     m  17.  [a  +  x)  (b-\-x)  = 
m-^n    "^  m  —  n  (c-^x){d-Yx) 


130  ELEMENTARY  ALGEBRA, 

Miscellaneous  Examples. 

EXERCISE    71. 


1. 

ve  : 

x—1      x—2      x—d      ^ 
6       '       9             12     ~ 

2. 

10a;-14      6X-4:      x-5 

3                   7       "■      2 

3. 

a;  — «      %x  —  b      a  —  x      10«  +  11J 
9               5                6     ~          3 

4. 

6a:  +  13      9a;  +  15       2x 

5                  5        "iVs 

5. 

10a;  +  17      36X  +  Q  _5x-4: 
6             11  a; -8"       3 

6. 

18^+r      ^^^     2.-1^  _3, 

7. 

«^      bm-^-n       1 
2a;"'        2         '  2a; 

8.  (2a;-l)(3a;  +  2)  =  (3a;-5)(2a;  +  20) 

9.  (3-a;)2-(a;-5)2=-4a; 

2a;  +  lV2      (2V5)a:-l_a:-% 

5  ^-Vs      "    2% 

10 -4a;       3 -2a; 
11 


12 


a;  +  l         Vs^H-^ 
a-^  —  a;         a;^  +  a;  3  a; 


a;3_2a;      x^-\-2x      o?  -  4.x 

13.  (6  «  -  2  a;)  (3  a  +  6  a;)  =  (10  a  +  2  a;)  (3  «  -  6  a;) 

l-\-2ax      4a;  — 2 

14.  a;  =  — ! 2 — 

15.  -07  (10a; +  1-3)= -3  (-43;- -01) +  -5 

.1 4.4.  r '11 

16.  -66  a; --^^^^^- — ^^  = -02  a;  + '198 


EQUATIONS  OF  ONE  UNKNOWN  QUANTITY.      131 
9  6  3 

17.  :; :;— i = '' T 

1—X       l-\-x  x-  —  1 

18.  {ex  -hf  =  Iy^  -  ah  '\-  {a  -  cxf 

4      ,   ,11   ,  5a:  +  20      12      ,  ^11 
^^•5^  +  ^2-5  +  9^=06  =  15^  +  ^25 

20.  a;  —  1  +  —  (2;  —  2)  =  -^ 
a^  '       or 

X  —  a  _a^  -\-V^      x  —  h 
h     ~     ah  a 

1-2 a; --06      -32 a;- '024  _      2-3 a; -1-8 
1-4  -35         ""  2 


22. 


a^  —  ^hx      ,„      hx  ,  Qhx  —  ^a^      hx4-4:a 

23.  X 5 h-  = ^r-5 z 

a^  a    ^         %a^  4a 

18a:+10  ,  16a;-14      72a;-f-30  ,  2OV2 
42       ^  18a:  +  6  108       ^   42 

1.1  1.1 

26- S  + 


a;  —  2      a;  —  5      a;  —  4      a;  —  3 


Examples  involving  Simple  Equations  of  One 
Unknown  Quantity. 

EXERCISE    72, 

A  has  twice  as  much  money  as  B,  and  C  has  three 
times  as  much  as  B,  and  they  together  have  $1200.  How 
much  has  each  ? 

Let  X  —  the  number  of  dollars  B  has, 

then  2  a;  =  the  number  A  has, 

and  3  a;  =  the  number  C  has. 

.*.  a;  +  2a;  +  3a;  =  1200  ,y 

6a;  =  1200 
X  =  200,  B's  number  of  dollars ; 

2  a;  =  400,  A's  number  of  dollars ; 

3  a;  =  600,  C's  number  of  dollars. 

1.  A  number  increased  by  3  times  itself  and  4  times 
itself  equals  80.     What  is  the  number  ? 


132  ELEMENTARY  ALGEBRA. 

2.  A  man  paid  $255  for  a  horse,  cow,  and  pig ;  the  cow 
cost  10  times  as  much  as  the  pig,  and  the  horse  4  times  as 
much  as  the  cow.     Kequired  the  cost  of  each. 

3.  If  a  certain  number  be  increased  by  the  sum  of  Y3 
and  Ye  of  itself  it  will  be  45.     What  is  the  number  ? 

4.  Grandfather's  age  diminished  by  its  Y4,  and  the  re- 
mainder diminished  by  its  Ys*  is  40  years.  What  is  his 
age? 

5.  A  farmer  raised  660  bushels  of  wheat  in  three  fields  ; 
lie  raised  Y3  as  much  in  the  second  field  as  in  the  first, 
and  Y5  as  much  in  the  third  as  in  the  second.  How  much 
did  he  raise  in  each  ? 

6.  A  number  increased  by  its  Ye^  and  the  result  dimin- 
ished by  its  %,  leaves  a  remainder  of  48.  What  is  the 
number  ? 

The  sum  of  two  numbers  is  70,  and  one  is  16  more  than 
the  other.     Eequired  the  numbers. 

Let  X  =  the  smaller  number, 

then     X  +  1Q  =  the  greater  number, 
and   2x  +  16  =  their  sura. 
.-.  2a;  +  16  =  70 
2a;  =  54 
x  =  27,  the  smaller  number ; 
a;  +  16  =  43,  the  greater  number. 

7.  The  sum  of  two  numbers  is  84,  and  the  greater  is 
12  more  than  double  the  smaller.     What  are  the  numbers  ? 

8.  A  horse  and  carriage  cost  $252,  and  the  cost  of  the 
horse  was  $12  more  than  twice  the  cost  of  the  carriage. 
Required  the  cost  of  each. 

9.  If  Ys  of  a  number  increased  by  10  more  than  Ys  of 
the  number  is  190,  what  is  the  number  ? 

10.  An  estate  of  $16,000  is  to  be  divided  among  A,  B, 
and  0.  B  is  to  have  $1000  more  than  4  times  as  much 
as  A,  and  C  is  to  have  $500  more  than  Y2  as  much  as  B. 
What  is  the  share  of  each  ? 


EQUATIONS  OF  ONE  UNKNOWN  QUANTITY.      I33 

11.  The  sum  of  two  numbers  is  163,  and  their  differ- 
ence is  19.     What  are  the  numbers  ? 

12.  A  man  has  three  coils  of  rope  containing  200  yards  : 
the  second  contains  73  as  much  as  the  first,  +  15  yards ; 
and  the  tliird  Ya  as  much  as  the  second,  + 10  yards.  How 
many  yards  are  there  in  each  coil  ? 


If  a  certain  number  be  doubled,  and  then  increased  by 
84,  it  will  be  five  times  the  number.    What  is  the  number  ? 
Let  X  =  the  number, 

then  2  a;  +  84  =  twice  the  number  increased  by  84, 
and  5  a;  =  five  times  the  number. 

.-.    2a; +  84  =  5a:  (Ax.  1) 
3a:  =  84 
a:  =  28 

13.  In  a  certain  orchard  one  half  the  trees  bear  apples, 
one  fourth  plums,  one  fifth  peaches,  and  the  remaining  20 
cherries.     How  many  trees  in  the  orchard  ? 

14.  A  man  spent  ^5  of  his  money  and  then  earned  $12, 
after  which  he  had  Y5  as  much  as  he  had  at  first.  How 
much  had  he  at  first  ? 

15.  Three  men  purchased  a  ship.  A  paid  Ygo  of  it ;  B, 
726  of  it ;  and  C,  the  remainder,  which  was  16600.  Ee- 
quired  tlfe  value  of  the  ship. 

16.  A  baker  uses  Y2  of  a  barrel  of  flour,  —  2  pounds,  at 
one  time  ;  74  of  the  remainder,  +  5  pounds,  at  another 
time  ;  and  Ys  of  what  then  remains,  +  4  pounds,  at  an- 
other time,  and  finds  the  barrel  empty.  How  many  pounds 
were  in  the  barrel  at  first  ? 

17.  If  a  certain  number  be  halved,  and  then  diminished 
by  14,  it  will  be  Vs  of  its  original  self.     Find  the  number. 

18.  A  certain  sum  is  to  be  divided  among  A,  B,  and  C. 
A  is  to  have  130  less  than  Y2  of  it,  B  $10  less  than  Ys 
of  it,  and  C  $8  more  than  Y4  of  it.  What  will  each 
receive  ? 

7 


13 i  ELEMENTARY  ALOEBRA. 

The  sum  of  two  numbers  is  121,  and  4  times  the  greater 
equals  7  times  the  less.     Required  the  numbers. 

Since  4  times  the  greater  equals  7  times  the  less,  the  greater 
equals  "^  j^  of  the  less. 

Let      4  a;  =  the  less, 
then     lx  =  the  greater, 
and    11  a;  =  121,  their  sum. 
a;  =  11 
4 a;  =  44,  the  less; 
7  a;  =  77,  the  greater. 

19.  Four  times  John's  age  equals  5  times  William's,  and 
the  difference  of  their  ages  is  4  years.    What  are  their  ages  ? 

20.  Two  thirds  of  A's  money  equals  Yg  of  B's,  and  they 
together  have  $5500.     How  much  has  each  ? 

21.  A  farmer  raised  500  bushels  of  corn  and  oats  ;  75  of 
the  quantity  of  corn  equaled  7io  of  the  quantity  of  oats. 
How  much  of  each  kind  did  he  raise  ? 

22.  A  and  B  together  husked  180  shocks  of  corn  ;  ^4  of 
the  number  A  husked  equals  7  more  than  Vig  of  the  num- 
ber B  husked.     How  many  did  each  husk  ? 


A  sold  a  horse  for  $160.50,  and  thereby  gained  7^. 
What  did  he  pay  for  the  horse  ? 

Let  X  =  the  cost  of  the  horse. 

7  7 

Since  he  gained  7^  he  gained  :r^  of  the  cost,  which  is  jttt  of  x, 

Ix 
or  ^  „  ,^ .  7  a/ 

^"^  Then  x  +  zttzt;,  =  the  selling-price. 

.'.    a;  +  ^  =  $160.50 
whence  a;  =  $150 

23.  A  miller  sells  corn  at  QQ  cents  a  bushel,  and  thereby 
gains  10  ^.     What  is  the  cost  of  the  corn  ? 

24.  If  a  number  be  increased  by  33  Vs^  of  itself,  and 
that  sum  increased  by  25  ^  of  itself,  it  will  be  110.  What 
is  the  number  ? 


EQUATIONS  OF  ONE   UNKNOWN  QUANTITY.      I35 

25.  A  farmer  lost  8  ^  by  selling  a  cow  for  $69.  What 
was  the  cost  of  the  cow  ? 

26.  A  and  B  together  have  $22,500,  and  A  has  25^ 
more  than  B.     How  much  has  each  ? 

27.  If  A  spends  25  ^  of  his  money,  and  then  earns  40  ^ 
of  what  he  has  remaining,  and  then  has  $140,  how  much 
money  has  he  at  first  ? 

28.  A's  house  cost  $300  more  than  B's,  and  24^  of  the 
cost  of  A's  equals  347,^  of  the  cost  of  B's.  What  was 
the  cost  of  each  ? 

29.  A  merchant  bought  cloth  20  ^  below  marked  price 
and  sold  it  at  a  gain  of  30  j^,  and  gained  70  cents  a  yard. 
What  was  the  marked  price  ? 


A  jeweler  bought  watches  for  $40  apiece  and  sold  them 
at  an  advance  of  $16  apiece.     What  was  his  gain  per  cent  ? 
Let  X  =  his  gain  per  cent ; 

then  rrnr  of  40,  or  -^  =  the  gain  vx  dollars ; 

whence  -=-  =  16 

0 

and  a;  =  40 

30.  Bought  a  cow  for  $50,  and  sold  her  for  $65.  What 
was  the  gain  per  cent  ? 

31.  Sold  a  horse  for  $150,  and  thereby  lost  $25.  What 
was  the  loss  per  cent  ? 

32.  By  what  per  cent  must  the  fraction  Vg  be  increased 
to  make  the  fraction  "/ig  ? 

33.  A  man  doubled  his  capital  each  year  for  three  years, 
and  the  fourth  year  lost  all  he  had  previously  gained. 
What  per  cent  did  he  lose  the  fourth  year  ? 

34.  A  sold  B  an  acre  of  land  for  $150,  and  gained  25  ^ 
of  the  cost ;  B  sold  it  to  C  for  the  same  price  that  A  paid. 
What  per  cent  did  B  lose  ? 


136  ELEMENTARY  ALGEBRA. 

What  principal  will  in  5  years  at  6^  amount  to  $624  ? 

Let  X  =  the  principal. 

30  3 

For  5  years  at  6  %,  j^  or  ^^  of  the  principal  equals  the  interest ; 

hence,  r^  of  a;,  or  ^^  =  the  interest ; 

whence  a;  +  j^  =  the  amount  in  dollars, 

and  a;  +  ~  =  024 

a:  =  480 

35.  What  sum  of  money  must  be  put  at  simple  interest 
for  8  years  at  472^  to  amount  to  $6800  ? 

36.  The  interest  on  a  certain  principal  for  7  years  at  6^ 
is  1464  less  than  the  principal.     Required  the  principal. 

37.  A  has  1200  more  than  B,  and  the  sum  of  their  in- 
terests for  5  years  at  4^  is  $160.  What  is  the  principal 
of  each  ? 

38.  A  gives  B  $800  for  3  years  at  6  ^  per  annum.  How 
many  dollars  must  B  give  A  for  4  years  at  5  ^  per  annum 
to  yield  the  same  amount  of  interest  ? 


In  how  many  years  will  $700  at  6^,  simple  interest, 
amount  to  $910  ? 

Let  X  =  the  number  of  years. 

R  q 

At  Q%  for  1  year,  r^  or  ^  of  the  principal  equals  the  interest,  and 

Sx 
for  X  years  the  interest  is  ^  of  the  principal ; 

hence  -^  of  700  =  the  interest  in  dollars ; 
50 

and      910  —  700  =  the  interest  in  dollars. 

Sx 


X  700  =  910  -  700  (Ax.  1) 
oO 


whence 


39.  In  what  time  will  $800  at  4V2^  amount  to  $1004, 
simple  interest  ? 


EQUATIONS  OF  ONE   UNKNOWN  QUANTITY.      137 

40.  In  what  time  will  $750,  at  673^  per  annum,  double 
itself  at  simple  interest  ? 

41.  In  what  time  will  a  dollars  at  r  per  cent,  simple 
interest,  treble  itself  ? 

42.  At  what  rate  will  $750  in  6  years,  at  simple  inter- 
est, amount  to  $1020  ? 

43.  At  what  rate  will  m  dollars  in  n  years,  at  simple 
interest,  double  itself  ? 

I  bought  a  $100  bond,  bearing  5^  interest,  for  $80. 
What  per  cent  of  my  investment  did  I  gain  annually  ? 
Let  X  =  the  annual  gain  per  cent, 

then     jr^  of  $80  =  the  entire  gain ; 

but  jttt:  of  $100  =  the  entire  gain, 
xuu 

whence  re  =  6^4 

44.  Bought  railroad  stock,  par  value  $50  a  share,  for 
$45  a  share;  the  company  declared  a  dividend  of  6^. 
What  per  cent  did  I  receive  on  my  investment  ? 

45.  At  what  price  must  I  buy  railroad  stock,  par  value 
$100  a  share,  in  order  that  a  6  ^  dividend  will  bring  me  an 
income  of  8^  on  my  investment  ? 

46.  I  bought  a  $50  share  for  $40 ;  the  company  de- 
clared a  dividend  which  I  found  was  '7^2^  of  diJ  invest- 
ment.    What  per  cent  of  the  par  value  was  it  ? 

47.  If  25^  of  the  par  value  of  stock  equals  40^  of  the 
market  value,  what  is  the  par  value  of  stock  that  is  selling 
at  $6272  a  share? 

"^  48.  If  stock  bought  at  90  yields  an  income  of  5^,  at 
what  price  would  it  yield  6  ^  ? 

49.  What  capital  invested  in  5's  at  80  will  yield  the 
same  income  as  $4500  invested  in  6's  at  90  ? 


138  ELEMENTARY  ALGEBRA, 

How  far  may  a  person  ride  in  a  coach,  going  at  the  rate 
of  5  miles  an  hour,  that  he  may  walk  back  at  the  rate  of 
2  miles  an  hour  and  be  gone  5  hours  ? 

Let  X  =  the  number  of  miles, 

then  will        -^  =  the  time  going ; 

and  -^  =  the  time  returning ; 

whence  -?  +  Tr  =  5 
5       2 

and  x  =  7^/^ 

50.  If  a  boat  sailed  down  a  stream  at  the  rate  of  10  miles 
an  hour  and  returned  at  the  rate  of  6  miles  an  hour,  and 
was  gone  6  hours,  how  far  did  it  sail  down  the  stream  ? 

61.  A  boat  whose  rate  of  sailing  in  still  water  is  10  miles 
an  hour,  goes  down  a  stream  whose  rate  is  two  miles  an 
hour,  and  returns,  making  the  round  trip  in  5  hours. 
How  far  does  it  go  down  the  stream  ? 

52.  A  boat  whose  rate  of  sailing  in  still  water  is  6  miles 
an  hour,  goes  a  miles  down  the  stream  in  one  half  the  time 
it  requires  to  return.     What  is  the  rate  of  the  current  ? 


A  can  do  a  piece  of  work  in  5  days  and  B  can  do  it  in 
8  days.     In  what  time  can  they  do  it  working  together  ? 
Let       X  =  the  number  of  days  required, 
then    —  =  the  part  they  can  do  in  1  day, 

-^  =  the  part  A  can  do  in  1  day, 
o 

■^  =  the  part  B  can  do  in  1  day, 

hence  —  =  -f  +  "o 
X      5       S 

53.  A  can  do  a  piece  of  work  in  4  days,  B  in  5  days, 
and  C  in  6  days.  In  what  time  can  they  do  it  working 
together  ? 


EQUATIONS  OF  ONE   UNKNOWN  QUANTITY.      I39 

64.  Two  pipes  can  fill  a  cistern  in  5  hours,  and  one 
alone  can  fill  it  in  8  hours.  In  what  time  can  the  other 
fill  it  ? 

55.  There  are  3  pipes  connected  with  a  reservoir :  the 
first  can  fill  it  in  10  hours,  the  second  in  8  hours,  and  the 
third  can  empty  it  in  6  hours.  In  what  time  will  it  be 
filled  if  all  run  together  ? 

56.  A  can  do  a  piece  of  work  in  2V2  days,  working  8 
hours  a  day,  and  B  can  do  it  in  SYa  days,  working  9  hours 
a  day.  In  how  many  days,  working  6  hours  a  day,  can 
they  together  do  it  ? 

57.  A  has  $800  and  B  has  Yg  as  much.  How  much 
must  A  give  to  B  in  order  that  A  may  have  %  as  much 
as  B? 

58.  B  has  $300  more  than  A,  and  earns  $5  a  day ;  A 
earns  $8  a  day.  How  much  must  each  earn  in  order  that 
they  may  have  the  same  sum  ? 

59.  A  man  has  two  horses,  and  a  saddle  worth  $10. 
The  first  horse  and  saddle  are  worth  %  as  much  as  the 
second  horse,  and  the  second  horse  and  saddle  ^Vzo  as  much 
as  the  first  horse.     Kequired  the  value  of  each  horse. 

60.  A  general  draws  up  his  army  in  the  form  of  a  square, 
and  has  140  men  over  ;  he  then  endeavors  to  increase  each 
side  by  2  men,  and  finds  he  lacks  24  men  to  complete  the 
square.     How  many  men  has  he  ? 

61.  A  is  15  years  old  and  B  is  30.  In  how  many  years 
will  Vs  of  A's  age  equal  Ya  of  B's  ? 

62.  A  man  loaned  $1500,  a  part  at  5j^  and  the  rest  at 
Q%  ;  his  annual  interest  was  $81.  How  much  did  he  loan 
at  5^? 

63.  How  many  pounds  of  sugar  at  10  cents  a  pound 
must  be  mixed  with  25  pounds  worth  8  cents  a  pound  to 
make  a  sugar  worth  874  cents  a  pound  ? 


140  ELEMENTARY  ALGEBRA. 

64.  A  man  agreed  to  work  one  year  for  $180  and  house- 
rent  free.  At  the  expiration  of  9  months  he  was  deprived 
of  work  by  sickness  for  the  rest  of  the  year,  but  retained 
the  house ;  he  was  paid  $120  in  money  for  his  services. 
What  was  the  house-rent  valued  at  ? 

65.  What  time  of  day  is  it  when  %  of  the  time  past 
noon  equals  ^/i  of  the  time  to  midnight  ? 

Suggestion. — Let  x  =  the  number  of  hours  past  noon. 

66.  At  what  time  of  day  is  the  time  past  noon  Y^  of  the 
time  past  midnight  ? 

67.  At  what  time  between  4  and  5  o'clock  are  the  min- 
ute- and  hour-hands  of  a  clock  together  ?  At  right  angles  ? 
Opposite  each  other  ? 

Suggestion. — At  4  o'clock  the  minute-hand  must  gain  20  minute- 
spaces,  5  or  35  minute-spaces,  and  50  minute-spaces  respectively. 

68.  A  son's  age  is  ^5  t^at  of  his  father's,  but  in  16  years 
it  will  be  %  that  of  the  father's.     What  are  the  ages  now  ? 

69.  A  and  B  together  can  do  a  piece  of  work  in  24  days, 
A  and  0  in  30  days,  B  and  C  in  40  days.  In  what  time 
can  they  do  it  all  working  together  ? 

70.  A  boy  spent  Yg  his  money  and  Y2  a  cent ;  then,  Y2 
of  the  remainder  and  Ys  a  cent ;  then,  Y2  of  what  then 
remained  and  Y2  a  cent,  and  had  9  cents  remaining.  How 
much  money  had  he  at  first  ? 


Simple  Equations  of  Two  Unknown  Quantities. 

Definitions  and   Principles. 

161.  A  single  equation  of  two  unknown  quantities  may 
be  satisfied  by  any  number  of  values  of  the  unknown  quan- 
tities, and  is  therefore  said  to  be  Indeterminate. 

Thus,  2  :r  —  ?/  =  10  is  true  when  x=z6  and  y  =  2  ; 
when  a;  =  7  and  ^  =  4  ;  when  x  =  8  and  ^  =  6  ;  etc. 


ELIMINATION  BY  SUBSTITUTION,  141 

162.  Two  simultaneous  simple  equations  of  two  un- 
known quantities  can  be  satisfied  by  only  one  pair  of  values 
of  the  unknown  quantities. 

Thus,  x-\-2y  =-1  and  bx  —  ^y  =  ^  are  satisfied  only 
by  a;  =  3  and  y  =  2. 

163.  Generally,  when  there  are  as  many  independent 
simultaneous  equations  given  as  there  are  unknown  quan- 
tities involved,  their  solution  can  be  effected  by  elimina- 
tion.    (See  page  96.) 

164.  There  are  three  easy  methods  of  elimination  : 

1.  By  addition  and  subtraction.      2.  By  substitution. 

3.  By  comparison. 

Note. — For  elimination  by  addition  and  subtraction,  see  pages  59 
and  96. 


Elimination  by  Substitution. 
Illustration. — 

Given  -!  _      ,  r^  \t^I  r  ^o  find  the  values  of  x  and  y. 

\Sx-{-    y  =  9  (B)  i  ^ 

Solution:  Transpose  3 a;  in  (B), 

y  =  9-Sx  (1) 

Substitute  (1)  in  (A), 

5a;  _  2(9 -3a;)  =  4  (2) 

Solve  (2),  a:  =  2 

Substitute  2  for  a;  in  (1),  and  reduce, 

y  =  d 

EXERCISE    73. 

Solve  : 


1. 


(4a;+7y  =  19)  4.    {6x-9y  =  0) 

I  3 a; +  2?/  =  11  )  (4a:  +  3y  =  3  j 

{3x-5y=    7)  5.    iSx-\-7y  =  SS) 

(4a;+7y  =  23i  \5x-Sy  =  15) 

{3x-2y=    5)  6.    {'7x-2y  =  27) 

\2x-\-3y  =  25]  (9a;  +  6i/  =  69) 


142  ELEMENTARY  ALGEBRA. 


i    Sx-by=  —  %)  10.    j242:-18^=    48) 
(10a:+7y=96     j  (  142;  + 24?/  =  166  f 

{llx-'iij=-l    I  11.    ja;  +  y  =  0  ) 

(12a;  +  9?/=-51  j  l25^  +  24?/  =  4f 

j  13a;-15«/=-13)  12.    (18^-15?/=    7) 
(14:  ic  +  17  ?/  =  -  14  f  (  15  a;  +  24 1/  =  91  f 


(1) 
(2) 


Elimination  by  Comparison. 

IUustration.-Solye  ]  ^ ^  +  f ^  =    ^^  t\\ 

Solution :  Transpose  5  y  in  (A)  and  divide  by  3, 

21-% 
^-^3~ 
Transpose  —  3y  in  (B)  and  divide  by  4, 

Compare  (1)  and  (2), 

21-5y_3y-l 
3-4 
Reduce  (3),  3/  =  3 

Substitute  3  for  y  in  (1)  and  reduce, 

EXERCISE    74. 

Solve  : 

1.  (     x-{-^y  =  10\  6.    )8.^'-    9y=:-66) 
|3a;  +  42/  =  24f  |7a;  +  10?/=    121  f 

2.  i4a;~22/  =  12)  7.  j2a;  +  3?/=    5) 
(3a;  +  2y=2!  (82;  +  9?/  =  18) 

3.  (6a;--5y=    4)  8.    il2a;-25?/=    1) 
(  4:^+7  2/  =  44  f  (22a;  +  15i/=14) 

4.  (3a;  +  5^=-    5)  9.    {l%x-1y  =  l\ 
(52;+3i/=-19)  1llic-5?/  =  8f 


5. 


(3^;-    ^  =  19)  10.    (  6a;  +  9?/  =  5  ) 

(     2;-3?/=    1  )  (9ic+6?/  =  5  f 


SOLUTION  OF  NUMERICAL  EQUATIONS,        143 


Solution  of  Numerical  Equations. 

2     "^      3 
a:-3   ,  y  +  4 


Ulugtration.— Solve 


=  7 
4 


(A) 
(B) 


Solation:  Clearing  (A)  and  (B)  of  fractions, 

33;  +  12  +  2y-   4  =  43  (1) 

4a; -12 +  32/ +  13  =  48  (2) 

Transposing  and  uniting  terms  in  (1)  and  (2), 

3a;  +  2y  =  34  (3) 

4a; +  32/ =  48  (4) 

Multiplying  (3)  by  3  and  (4)  by  3, 

9a; +  63/ =  103  (5) 

8a; +  62/=   96  (6) 

Subtracting  (6)  from  (5), 

x  =  Q 
Substituting  6  for  x  in  (3)  and  reducing, 
2/  =  8 


Solve  : 


1. 


3. 


6. 


^  +  y3^  =  24 


5      , 


EXERCISE    73. 
2. 


24 


x-\-^y      3a;-2y_36 
5       "^        6        ~  5 

2a;  +  4y      5a:  +  4y_ 
16  31 


7ir  —  5y  =  6 

|(=^  +  7)-|(y-9)-3 


3         5  . 


3      ,   2 

10^+3^' 


11 


12  +  18-    ^ 

i;+i^=io 


144 


ELEMENTARY  ALGEBRA, 


8. 


10. 


Sx  —  y      4:X-\-2y 

-4 

6                11       ~ 

5a;  +  32/      2x-ly 

-20 
J 

[       10         '          3        ~ 

2                 3 
x-^'-Zy      x  —  'dy  - 

^-    ^x-\-y      5 
. x-y-3 

x-{-20y  =  4oO 

3a;  — 4^  = 

'  5a;  +  6y      Sx  —  4:y 

5  3 

l!x-{-9y  _9x-\-8y 

11       ""       12 


12 


=  20  J 


11. 


1  _  ^+1  =    ^^    ' 
X  —  y      X  —  y 

23        -^ 


12. 


12^  +  7a; 

bx  —  'dy 
l^y-'ix 

6a;  +  10 


=  1 

=  2 


13. 


a;  —  3y      '^^-\-^y  _  _. 

9~^  43        ~  ~ 

5  ?/  —  7  a:  ^ 


13 


14. 


3x  —  Sy      6x  —  Sy 

f""  14 


14 


^  +  ^y_. 


11 


2a;  +  ^  +  48 


15.    (8x  —  5y      lly  —  4:X_ 

7     '  "^  5         ~ 

17^j-j^      2^_ 
5  "^  3   ~^ 


16. 


2 


SOLUTION  OF  NUMERICAL  EQUATIONS.       145 


165.  It  is  sometimes  easier  to  eliminate  one  of  the  un- 
known quantities  without  clearing  of  fractions. 


Illustration. — Solve 


2a;  "^32^       6 


(A) 
(B) 


6 3^_13 

Solution :   Reduce  the  corresponding  terms  in  (A)  and  (B)  to  a 

common  denominator,      9         \q       §4 

65"*"  ny'^n 

10        9        13 

6a;      123/ ~  72 

Multiply  (1)  by  10  and  (2)  by  9, 

90       160  _  840 

6a;  "^  12y~  72 

90  _  81       117 

6a;      122/ ~  72 

Subtract  (4)  from  (3),         241  _  723 

\2y~  72 

Clear  of  fractions,    723  x  12y  =  72  x  241 

72  X  241 


(1) 
(2) 

(3) 
(4) 

(5) 


Substitute  2  for  y  in  (A), 


^  ~  723  X  12 


=  2 


3^      i_I 
la;"^  6  ~  6 

A_i 

2a;~  6 

6a;  =  18 

a;  =  3 


EXERCISE    76. 


Solve  : 
1. 


1,1    . 

X      y 
X      y 


x^l ' y+1 
La;+l^y-f  1 


=  2 


3       2^131 

x^  y~  VZ 

5       7^29 
x'^  y      12 


Zx^^y        4 

A.  4.  1_  =  75. 
dx'^2y        6J 


[46 

ELEMENTARY  ALGEBRA 

V 

5. 

M     1      ^        rl 

7. 

x+2  ' y-^"^ 
3             1          1   ■ 

U  +  2    y-^     2j 

3         5   _       fil 

2a;      2y-      "2  J 

6. 

^      '       1       ^         5^ 

8. 

|i  +  4  =  3*    1 

2(a;  +  l)  '  3(y+l)~ 

^                   1             1 

2         4 

.^(^  +  1)      3(^  +  1)"     J 

[32;      6y~       ^J 

Solution  of  Literal  Equations. 

IUustrations.-SolYe  J  «  ^  +  %  =  ^       ( A) ) 
\mx-{-ny  =  d       (B)  f 

Solution:  Multiply  (A)  by  m  and 

(B)  by  a, 

amx  +  bmy  =  cm 

(1) 

amx  +  any  —ad 

(2) 

Subtract  (2)  from  (1), 

(Jm  —  a7i)y  =  cm 

-ad              (3) 

cm 

-a<^ 

y=i^, 

—  an 

Multiply  (A)  by  n  and  (B)  by  h. 

anx  +  hny  =  en 

(4) 

hmx  +  hny  =z  hd 

(5) 

Subtract  (5)  from  (4), 

{an  —  'bm)x  =  cn 

-J<^ 

en 

-^>^ 

x  = 

an 

—  bm 

a   ,    h 

^     y 

(A)' 

Solve  - 

a       c       ^ 

U-y  =  ' 

(B) 

Solution:  Subtract  (B)  from  (A), 

b  +  c 

y    -"-' 

(1) 

b  +  c=z(c  —  d) 

y         (2) 

b  +  c 

y  =  ._^ 

SOLUTION  OF  LITERAL  EQUATIONS. 


147 


EXERCISE     77. 


Solve 

1 


iax-\-hy  =  c) 
ax  —  by  =  d\ 

{x-jry  =  m      ) 
\  ax-\-hy  =  n  ) 


^  ,  y 

a      0 


1=^' 


j  mx-\-ny=p 
\  rx-\-sy  =  q 


a   ,   b 
X      y 

X       y 


j  ax 
\bx 


+  my 
-my 


'•! 


X  —  y  =  a 
mx-{-ny 


10. 


X 

+  ^.= 

1 

a 

a- 

a 

X 

y  - 

1 

b 

w~ 

b 

{cx  —  dy  =  b   ) 
(  mx  —  ny=-  b  \ 


a 
mx 

+ 

b  __ 
ny~ 

c 

a 
nx 

+ 

b   _ 
my 

d 

Examples  involving  Simple  Equations  of  Two 
Unknown  Quantities. 

EXERCISE    78. 

1.  A  man  bought  two  farms,  one  of  80  acres  and  one 
of  50  acres,  for  $22190.  Had  the  first  contained  70  acres 
and  the  second  60  acres,  the  second  would  have  cost  ^%8 
as  much  as  the  first.  What  was  the  price  of  each  farm 
per  acre  ? 

2.  Two  thirds  of  A's  fortune  added  to  three  fourths  of 
B's  is  $700,  and  B's  increased  by  $100  is  five  sixths  of  A's. 
What  is  the  fortune  of  each  ? 

3.  The  sum  of  two  numbers  is  7,  and  if  the  larger  be 
added  to  the  numerator  and  the  smaller  to  the  denominator 
of  the  fraction  Y^,  the  result  wiU  equal  Y*.  What  are  the 
numbers  ? 


148  ELEMENTARY  ALGEBRA, 

4.  The  sum  of  two  fractions  is  ^Yig,  and  their  difference 
is  ^Yie .     What  are  the  fractions  ? 

5.  A  man  has  a  certain  quantity  of  oats  and  corn.  If 
he  mixes  two  thirds  of  his  oats  with  one  half  of  his  corn, 
he  will  have  a  mixture  of  60  bushels  ;  bui)  if  he  mixes  all 
his  corn  with  four  fifths  of  his  oats,  the  oats  in  the  mixture 
will  exceed  the  corn  by  8  bushels.  How  many  bushels  of 
each  kind  has  he  ? 

6.  A  man  has  two  watches  and  a  chain.  The  first  watch 
is  worth  $60.  If  the  chain  be  put  on  the  first  watch  they 
together  will  be  worth  %  as  much  as  the  second ;  but  if 
the  chain  be  put  on  the  second  watch  they  together  will 
be  worth  twice  as  much  as  the  first.  Required  the  value 
of  the  second  watch  and  chain  respectively. 

7.  If  2  be  added  to  the  numerator  of  a  fraction  it  will 
be  V2,  but  if  3  be  added  to  the  denominator  it  will  be  Vs. 
What  is  the  fraction  ? 


Let  —  =  the  fraction ;  then, =  -^ ,  and :^  =  77 . 

y  '  '      2/  2'  y  +  S       3 

8.  If  3  be  added  to  both  terms  of  a  fraction  it  will  be 
%,  but  if  3  be  subtracted  from  both  terms  it  will  be  %. 
What  is  the  fraction  ? 

9.  The  difference  of  two  numbers  is  5,  and  if  the  greater 
be  subtracted  from  the  numerator,  and  the  less  from  the 
denominator  of  ^y^,  the  result  will  be  7?.  What  are  the 
numbers  ? 

10.  There  is  a  number  consisting  of  two  figures,  such 
that  if  9  be  added  to  the  number  the  figures  will  change 
places,  and  the  sum  of  the  figures  is  7.  Required  the 
number. 

Suggestion.— Let  x  =  the  ten's  figure  and  y  the  unit's  figure. 
Then,  lOx  +  y  =  the  number,  and  lOy  +  x,  the  number  with  the  fig- 
ures interchanged ;  whence, 

lOx  +  y  +  9  =  10y  +  x  (A) 

x  +  y  =  7  (B) 


EQUATIONS  OF  TWO   UNKNOWN  QUANTITIES.    149 

11.  The  sum  of  the  two  digits  of  a  number  is  12,  and 
if  54  be  added  to  the  number  the  digits  will  change  places. 
What  is  the  number  ? 

12.  A  certain  number  is  four  times  the  sum  of  its  two 
digits,  and  if  9  be  added  to  the  number  its  digits  will 
change  places.     Required  the  number. 

13.  The  difference  of  the  two  digits  of  a  number  is  Yig 
of  the  number,  and  if  6  be  added  to  the  number  its  value 
will  be  five  times  the  sum  of  the  digits  of  the  original 
number.     Required  the  original  number. 


14.  A  and  B  together  can  do  a  piece  of  work  in  8  days, 
and  A  can  do  as  much  in  3  days  as  B  can  do  in  5  days. 
In  how  many  days  can  each  alone  do  it  ? 

Suggestion. — Let  x  =  the  time  in  which  A  can  do  it, 
and  y  =  the  time  in  which  B  can  do  it. 

—  =  the  part  A  can  do  in  1  day. 

—  =  the  part  B  can  do  in  1  day. 

■^  =  the  part  both  can  do  in  1  day. 
Ill' 
Since  A  can  do  as  much  in  3  days  as  B  can  do  in  5  days, 

'^='-  (B) 

X       y  ^  ^ 

15.  If  A  works  3  days  and  B  5  days,  they  can  accom- 
plish a  piece  of  work ;  but  if  A  works  2  days  and  B  3  days, 
they  will  accomplish  only  %  ot  it.  In  what  time  can  each 
alone  do  it  ? 

16.  Two  thirds  of  what  A  can  do  in  a  day  equals  three 
fourths  of  what  B  can  do,  and  they  together  can  do  a  job 
in  8  days.     How  long  would  it  take  each  alone  to  do  it  ? 

17.  Five  men  and  3  boys  can  do  a  piece  of  work  in  6 
days,  and  4  men  can  do  as  much  as  6  boys.  Ilow  long 
would  it  take  1  man  and  1  boy  each  to  do  it  ? 


150  ELE3IENTARY  ALGEBRA. 

18.  A  field  may  be  divided  into  8  lots  of  one  size  and 
9  lots  of  another  size ;  but  4  lots  of  the  first  size  and  10 
of  the  second  size  together  will  occupy  only  ^1^  of  the 
field.  Into  how  many  lots  of  each  size  may  the  field  be 
divided  ? 

19.  The  distance  around  a  room  is  52  feet,  and  if  4  feet 
be  added  to  the  length  it  will  be  twice  the  width.  Ee- 
quired  the  length  and  width  respectively. 

Suggestion. — Let  x  =  the  length  and  y  the  width,  then 

2x  +  2y  =  52f  the  number  of  feet  around  the  room ; 
also,  X  +  4:  =  2y,  twice  the  width. 

20.  A  man  has  two  square  fields,  one  of  which  is  6  rods 
longer  than  the  other,  and  the  sum  of  the  distances  around 
them  is  96  rods.     What  is  the  length  of  each  field  ? 

21.  A  man  has  a  rectangular  lot,  such  that  twice  the 
length  increased  by  6  yards  equals  four  times  the  width 
diminished  by  4  yards,  and  the  distance  around  it  is  50 
yards.     Eequired  the  length  and  width  respectively. 

22.  A  rectangular  field  has  a  perimeter  of  52  rods,  and 
if  its  width  be  increased  by  6  rods  and  its  length  by  8  rods, 
the  width  will  be  %  of  the  length.  Required  the  dimen- 
sions of  the  rectangle. 

23.  A  certain  fishing-rod  consists  of  two  parts :  the 
length  of  the  upper  part  is  y^  of  the  length  of  the  lower 
part,  and  9  times  the  upper  part  together  with  13  times 
the  lower  part  exceed  11  times  the  whole  rod  by  36  inches. 
Find  the  length  of  the  two  parts. 

24.  A  and  B  ran  a  race  which  lasted  5  minutes :  B  had 
a  start  of  20  yards,  but  A  ran  3  yards  while  B  ran  2,  and 
won  by  30  yards.  Find  the  length  of  the  course  and  the 
speed  of  each. 

25.  A  man  having  worked  20  days  and  been  idle  8  days, 
saved  $50.  Had  he  worked  24  days  and  been  idle  12  days, 
he  would  have  saved  $57.  What  were  his  daily  wages, 
provided  he  maintained  himself  ? 


EQUATIONS  OF  TWO   UNKNOWN  QUANTITIES.    151 

26.  If  the  length  of  a  rectangle  be  increased  by  2  feet 
and  the  width  by  3  feet,  the  area  will  be  increased  by  42 
square  feet ;  but,  if  the  length  be  diminished  by  2  feet  and 
the  width  be  increased  by  4  feet,  the  area  will  be  increased 
by  12  feet.  Kequired  the  length  and  width  of  the  rect- 
angle. 

Suggestion. — Let  x  equal  the  length  and  y  equal  the  width. 

27.  If  a  farmer  had  planted  5  more  hills  of  com  in  one 
row,  and  had  planted  5  more  rows,  he  would  have  had  700 
hills  of  corn  more ;  but,  had  he  planted  5  hills  less  in  one 
row,  and  4  rows  less,  he  would  have  had  620  hills  less. 
How  many  hills  did  he  plant  ? 

28.  If  there  had  been  two  more  persons  at  a  dinner- 
party, and  each  person  had  paid  one  shilling  less,  the  entire 
bill  would  have  been  4  shillings  more ;  but  if  there  had 
been  two  persons  less,  and  each  person  would  have  paid 
two  shillings  more,  the  bill  would  have  been  2  shillings 
less.     Kequired  the  bill  and  number  of  persons. 

29.  If  the  length  of  a  rectangle  were  diminished  by  5 
feet  and  the  width  increased  by  4  feet,  the  area  would 
remain  unchanged  ;  but,  if  the  length  were  to  remain  un- 
changed and  the  width  increased  by  7  feet,  the  area  would 
be  increased  by  224  square  feet.  Required  the  dimensions 
and  area  of  the  rectangle. 

30.  A  certain  sum  of  money,  put  out  at  simple  interest, 
amounts  in  6  years  to  1780,  and  in  10  years  to  $900.  Re- 
quired the  sum  and  rate  per  cent. 

31.  A  certain  principal  in  a  given  time  at  3  per  cent 
amounts  to  1920,  and  at  5  per  cent  for  the  same  time  to 
$1000.     Required  the  principal  and  time. 

32.  If  two  trains  start  from  two  stations  40  miles  apart 
at  the  same  time,  and  approach  each  other,  they  will  meet 
in  one  hour ;  but  if  they  run  in  the  same  direction  it  will 
require  the  faster  train  4  hours  to  overtake  the  slower. 
What  are  their  respective  rates  of  running  ? 


152 


ELEMENTARY  ALGEBRA. 


33.  A  passenger-train  200  feet  long  will  pass  a  freight- 
train  680  feet  long  in  30  seconds,  if  they  run  in  opposite 
directions ;  hut  if  they  run  in  the  same  direction  it  will 
require  1  minute  to  pass  it.  How  many  miles  per  hour 
does  each  train  run  ? 

34.  A  and  B  jointly  loan  C  a  sum  of  money  which  in 
five  years  at  6  per  cent  amounts  to  11170  ;  60  per  cent  of 
A's  share  of  the  principal  equals  75  per  cent  of  B's  share. 
How  much  of  the  amount  belongs  to  each  ? 


Simple  Equations  of  Tiiree  or  more  Unknown 
Quantities. 

Illustrative  Examples. — 


1.  Solve 


2  +  3  +  4-^^ 

-  4-  ^  4-  -  =  47 
3^4^5 


4"^5"^6 


38 


Solution:  Clear  equations  A,  B,  and  C  of  fractions, 

Qx+    4y+    Sz=    744  (1) 

20rc  + 15y  + 132  =  2820  (2) 

15a; +  12y  + 102  =  2280  (3) 

Multiply  (1)  by  4  and  bring  down  (2), 

24  a; +  162/ +  122  =  2976  (4) 

20a: +  15?/ +  122  =  2820  (2) 

Subtract  (2)  from  (4),  4:X  +  y=   156  (5) 

Multiply  (1)  by  10  and  (8)  by  3, 

60a;  +  40y  +  302  =  7440  (6) 

45a; +  36y  + 302  =  6840  (7) 

Subtract  (7)  from  (6),       15 a; +  4^=600  (8) 

Multiply  (5)  by  4,  16  a; +  4  2/=   624  (9) 

Subtract  (8)  from  (9),  x=     24 

Substitute  value  of  x  in  (8),  and  reduce, 

y=     60 
Substitute  values  of  x  and  y  in  (1),  and  reduce, 

z=   120 


EQUATIONS  OF  THREE  UNKNOWN  QUANTITIES.    153 


[^  +  ^  +  ^  =  29 

(A)' 

2.  Solve  - 

^-i+l=  9 

(B) 

■ 

\x  '  y      z 

(C) 

Solution :  Add  (A)  and  (B), 

M=- 

(1) 

Multiply  (A)  by  4  and  (C)  by  3, 

(2) 

X      y      z 

(3) 

Subtract  (3)  from  (2),    -  1  +  ^  =  123 

X       z . 

(4) 

Multiply  (4)  by  7  and  bring  down  (1), 

X         z 

(5) 

1+1  =  3« 

X        z 

(1) 

Add  (5)  and  (1),                      ?^  =  893 

(6) 

Divide  by  223,                            i=     4; 

whence  z- 

1 
~  4 

Substitute  the  value  of  z  in  (1),  and  reduce, 

1 


Substitute  the  values  of  x  and  z  in  (A),  and  reduce, 

1 

2^=0 


(rr  +  y  =  14  (A) 

3.  Solve  \x^z  =  \^  (B) 

(y  +  2:=18  (C)^ 
Solution :  Take  the  sum  of  (A),  (B),  and  (C), 

2x  +  2y  +  2z  =  48  (1) 

Divide  (1)  by  2,       x  +  y  +  z  =  24:  (2) 
Subtract  (A)  from  (2),           z  =  10 
Subtract  (B)  from  (2),           y=   S 
Subtract  (C)  from  (2),           x=   Q 


154: 


ELEMENTARY  ALGEBRA. 


Solve  : 


EXERCISE    79. 


3x-27/  +  4:Z=  - 
4,x-\-2i/~5z=  - 

2x-\-4:y--5z=    6  I 
x-{-Sy  —  2z  =  10) 


3. 


1/ 

2x-^dy-z=l2  \ 
4:X  —  2y-{-z=    3  [• 

i^+iy  +  i.  =  23 


3^+2^+4^ 


25 


ia;  +  iy+l.  =  27 


=  m 
=  ?^ 
=  r 

15 
5 
3 

20 
34 
44 


=  12 


X-}-  z  = 


^  +  '=2 


X      y 
X       z 


11. 


12. 


13. 


1  1,1 

2^-4^+8^ 

j     a;-f-2?/  —  3^  = 

4a:  — 4?/—     z  = 

[  3x  +  8y-i-2z  = 

fx-i-y-{-z  =  12 
x-{-y-J^u  =  ll 

y-^z-\-tt=    9  J 

^  +  ^-^=.    8 

X      y       z 

a:      y  '   z 


14.  x-{-2y  =  5Z'-10x  —  y-{-z  =  60 

15.  r  «iC+^?/-C^=z:«2_|_^2_^2 

i       ax  —  h y  -\-  c z  =  a-  —  b'^  -{-  c^ 
(  —  ax  -\-l  y  -\-  c  z  =  h^  -\-  c^  —  a^ 


y      z 


EQUATIONS  OF  THREE  UNKNOWN  QUANTITIES,  155 

Examples  involving  Simple  Equations  of  Three 
or  More  Unknown  Quantities. 

EXERCISE    so. 

1.  If  5  bushels  of  corn,  6  bushels  of  oats,  and  8  bushels 
of  rye  together  are  worth  $10.30  ;  3  bushels  of  corn,  5 
bushels  of  oats,  and  8  bushels  of  rye,  $8. 75  ;  and  1  bushel 
of  oats  mixed  with  1  bushel  of  rye  is  worth  as  much  as  1% 
bushel  of  corn — what  is  the  value  of  each  per  bushel  ? 

2.  A's  farm,  plus  Y3  of  B's  and  C's,  equals  230  acres ; 
B's,  plus  Y4  of  A's  and  C's,  equals  A's ;  and  C's,  plus  Vg 
of  A's  and  B's,  equals  170  acres.  How  many  acres  are 
there  in  each  farm  ? 

3.  If  A  should  give  B  one  half  of  his  money,  and  then 
B  give  C  one  half  of  his,  C  would  have  $550  ;  if  B  should 
give  C  one  half  of  his  money,  and  then  C  give  A  one  half 
of  his,  A  would  have  $800  ;  if  C  should  give  A  one  half 
of  his  money,  and  then  A  give  B  one  half  of  his,  B  would 
have  $750.     How  much  has  each  ? 

4.  A,  B,  and  C  together  can  do  a  piece  of  work  in  5  7ii 
days  ;  A  can  do  twice  as  much  as  B  or  three  times  as  much 
as  C  in  a  day.     How  long  will  it  take  each  alone  to  do  it  ? 

5.  The  sum  of  A's  and  B's  ages  is  55  years ;  the  sum 
of  A's  and  C's  is  62  years ;  and  the  sum  of  B's  and  C's 
is  77  years.     Required  the  age  of  each. 

6.  A  and  B  can  do  a  piece  of  work  in  4  days,  A  and  C 
in  5  days,  and  B  and  C  in  6  days.  In  what  time  can  each 
alone  do  it  ? 

7.  Two  supply-pipes,  A  and  B,  and  one  discharge-pipe, 
C,  are  connected  with  a  cistern.  If  the  three  pipes  run 
together  for  2  hours,  the  cistern  will  be  Veo  full ;  if  A  runs 
3  hours,  B,  4  hours,  and  C,  2  hours,  it  will  be  V30  full  ; 
and  if  A  runs  5  hours,  B,  3  hours,  and  C,  2  hours,  it  will 
be  Yio  full.  How  long  will  it  take  A  and  B  each  to  fill  it, 
and  C  to  empty  it  ? 


156  ELEMENTARY  ALGEBRA. 

8.  A  man  bought  a  horse,  carriage,  and  harness  for 
1500.  The  horse  cost  15  more  than  the  carriage  and  har- 
ness, and  the  carriage  cost  Ys  as  much  as  the  horse  and 
harness.     Eequired  the  cost  of  each. 

9.  There  is  a  number  consisting  of  three  digits  :  the 
sum  of  the  digits  is  13  ;  the  middle  digit  is  Yg  of  the  other 
two ;  and  if  297  be  added  to  the  number  the  unit's  and 
hundred's  digits  will  change  places.    Eequired  the  number. 

10.  A's  money  in  9  years  at  6^  will  produce  as  much 
interest  as  the  sum  of  B's  and  O's  in  473  years  at  4^ ; 
B's  in  8  years  at  5  ^  as  much  as  A's  and  C's  in  3  Y3  years 
at  6fo ;  and  C's  in  7  years  at  3^,  $42  more  than  A's  and 
B's  in  3  years  at  4^.     Eequired  the  principal  of  each. 

11.  A,  B,  C,  and  D  received  $1000.  B  got  half  as  much 
as  A.  The  excess  of  C's  share  oyer  D's  was  Y3  of  A's  share, 
and  B's  share,  increased  by  $100,  was  equal  to  the  sum  of 
C's  and  D's  shares.     How  much  did  each  receive  ? 

12.  If  40  peaches  are  worth  as  much  as  20  quinces  and 
4  oranges  ;  and  40  quinces  are  worth  as  much  as  30  peaches 
and  12  oranges ;  and  40  oranges,  70  peaches,  and  20  quinces 
are  worth  $4 — what  is  the  price  of  each  apiece  ? 

13.  A  man  has  $180  in  three  parcels.  If  he  takes  $20 
from  the  first  parcel  and  puts  it  with  the  second,  and  then 
takes  one  half  of  the  second  and  puts  it  with  the  third,  the 
third  will  be  worth  twice  as  much  as  the  other  two  ;  but 
if  he  takes  $20  from  the  third  parcel  and  puts  it  with  the 
second,  and  then  takes  one  half  of  the  second  and  puts  it 
with  the  first,  the  value  of  the  first  will  be  %  of  the  value 
of  the  third.     Eequired  the  value  of  each  parcel. 

14.  If  5  casks,  3  cans,  and  2  jugs  of  oil  be  drawn  from 
a  barrel  containing  60  gallons,  it  will  remain  ^Yso  ^^11  ?  i^ 
4  casks,  5  cans,  and  8  jugs  be  drawn,  it  will  remain  Y20 
full ;  and  if  3  casks,  5  cans,  and  10  jugs  be  drawn,  it  will 
remain  Ye  f^li-  What  is  the  capacity  of  a  cask,  a  can,  and 
a  jug  respectively  ? 


GENERALIZATION  AND  SPECIALIZATION.     157 

Generalization  and  Specialization. 

1.  Definitions. 

166.  Any  question  proposed  for  solution  is  a  Problem. 

167.  A  problem  whose  given  quantities  are  literal,  or 
general,  is  a  general  problem. 

168.  A  problem  whose  given  quantities  are  numerical, 
or  special,  is  a  special  problem,  or  an  example, 

169.  A  number  of  examples  with  different  given  quanti- 
ties but  like  conditions  and  requirements  constitute  a  class. 

170.  A  general  problem  involves  a  whole  class  of  ex- 
amples. It  is  the  type  of  a  class,  and  its  solution  the  solu- 
tion of  a  class. 

171.  The  solution  of  a  general  problem  gives  rise  to  a 
formula,  which,  interpreted,  gives  a  rule  for  the  solution 
of  every  example  of  a  class. 

172.  The  process  of  converting  a  special  problem  into  a 
general  one,  by  substituting  literal  for  numerical  quanti- 
ties, is  Generalization. 

173.  The  process  of  converting  a  general  problem  into  a 
special  one,  by  substituting  numerical  for  literal  quanti- 
ties, is  Specialization. 

2.  Examples. 

Ulustrationg. — 1.  If  A  can  do  a  piece  of  work  in  4  days 
and  B  can  do  it  in  5  days,  in  what  time  can  they  do  it 
working  together  ?    Generalize  this  question  and  solve  it. 

Solution:  Put  a  for  4  and  b  for  5.  Let  x  equal  the  time  re- 
quired for  both  to  do  it. 

Then    i  +  ^  =  i  (A) 

,                      ab        4x5      20„2, 
whence,  x  — r  =  t — ~  =  tt  =  2  tt  days. 

8 


158  ELEMENTARY  ALGEBRA. 

2.  A  and  B  can  do  a  piece  of  work  in  a  days,  A  and  C 
in  h  days,  and  B  and  C  in  c  days ;  in  what  time  can  each 
alone  do  it  ?  Solve  this  problem  and  specialize  for  a  =  10, 
J  =  8,  and  c  =  6. 

Solution :  Let  x  equal  the  time  required  hy  A,y  the  time  required 
by  B,  and  z  the  time  required  by  C  ;  then 


X      y      a 

(A) 

X      z       h 

(B) 

1+1=1 

y     z      c 

(C) 

Adding  (A),  (B),  and 

L  (C),  and  subtracting  from  the  sum  twice  (A), 

twice  (B), 

and  twice 

(C) 

respectively,  we 

have 

2_ 

z  ~ 

ac  +  ah  —  he 
ahc 

(1) 

2  _ 

y~ 

ah  +  he  —  ac 
ahc 

(3) 

2_ 

X 

be  +  ac  —  ah 
ahc 

(3) 

whence 

x  = 

2ahc 

(a) 

'  be  +  ac  —  ab 

y  = 

2abc 

(*) 

' ab  +  be-  ae 

Z  — 

2abc 

(c) 

'ac  +  ab  —  ac 

Put  a 

=  10,  1  = 

:  8,  and  c  =  6, 

X- 

3  X  10  X  8  X  6 

=  34| 

■  48  +  60-80  " 

y  = 

2  X  10  X  8  X  6 

■  80  +  48-60  ■ 

-4t 

z  = 

2  X  10  X  8  X  6 

•  60  +  80-48  ■ 

=  -i 

EXERCISE    81. 

1.  The  sum  of  two  numbers  is  20,  and  their  difference 
is  8.     Find  the  numbers. 

Snggestion. — Generalize  by  putting  a  for  20  and  h  for  8  in  the 
problem,  then  20  for  a  and  8  for  b  in  the  result. 


GENERALIZATION  AND  SPECIALIZATION.     159 

2.  A's  age  is  three  times  B's,  but  in  12  years  it  will 
be  only  twice  B's.     Kequired  the  age  of  each. 

Suggestion. — Put  m  for  3,  n  for  2,  and  t  for  12  in  the  problem, 
and  3  for  m,  2  for  n,  and  12  for  <  in  the  result. 

3.  A  and  B  have  $170,  and  %  of  A's  share,  equals  Y4 
of  B's.     How  much  has  each  ? 

Suggestion. — Put  m  for  ^/g,  n  for  '/4,  and  c  for  170,  etc. 

4.  A  can  do  a  piece  of  work  in  6  days  and  B  can  do  it 
in  8  days.  In  what  time  can  they  do  it  working  together  ? 
Generalize  and  solve. 

5.  A  has  $400  more  than  B,  and  B  has  $500  less  than 
C,  and  they  together  have  $1800.  How  much  has  each  ? 
Generalize  and  solve. 

6.  If  a  certain  number  be  increased  by  20,  the  result 
will  be  twice  as  great  as  when  the  number  is  diminished 
by  10.     Required  the  number.     Generalize  and  solve. 

7.  What  number  added  to  both  terms  of  the  fraction  Y7 
will  give  the  fraction  ^4  ?    Generalize  and  solve. 

8uggestion.^Put  —  for  ^  and  —  for  -7-. 

8.  B  has  40  acres  more  land  than  A,  but  if  A  buys  60 
acres  from  B,  A  will  have  175  times  as  much  as  B.  How 
many  acres  has  each  ?    Generalize  and  solve. 

9.  If  a  man  work  5  days  and  a  boy  3  days,  they  to- 
gether earn  $23,  but  if  the  man  and  boy  each  work  4  days 
they  together  earn  $20.  Required  the  daily  wages  of  each. 
Generalize  and  solve. 

10.  The  sum  of  A's  and  B's  ages  is  c  years,  and  A  is 
d  years  older  than  B.  Required  the  age  of  each.  Special- 
ize by  making  c  =  3G  and  c?  =  8  in  the  result. 

11.  Mr.  Jones  has  a  coins  worth  a  dollar;  some  of 
them  are  c-cent  pieces,  and  the  rest  are  d-cent  pieces. 
How  many  of  each  are  there  ?  Specialize  by  making 
a  =  14,  c  =  10,  and  c?  =  5. 


160  ELEMENTARY  ALGEBRA. 

12.  The  sum  of  three  consecutive  numbers  is  18.  Re- 
quired the  numbers.     Generalize  and  solve. 

13.  James  is  a  years  younger  than  William  ;  but  if  m 
times  James's  age  be  subtracted  from  n  times  William's, 
the  remainder  will  be  d  years.  How  old  is  each  ?  Spe- 
cialize by  making  «  =  4,  m  =  2,  n  =  ^y  and  d  =  22. 

14.  If  a  cows  and  b  oxen  are  worth  m  dollars,  and  c 
cows  and  d  oxen,  n  dollars,  required  the  value  of  a  cow 
and  of  an  ox.  Specialize  by  putting  5  for  a,  7  for  b, 
10  for  c,  3  for  d,  370  for  m,  and  355  for  n, 

15.  A  and  B  can  do  a  piece  of  work  in  d  days.  After 
working  together  c  days,  B  leaves,  and  A  does  the  balance 
in  a  days.  In  what  time  could  each  do  it  alone  ?  Special- 
ize by  putting  30  for  d,  18  for  c,  and  20  for  a. 

16.  If  a  certain  rectangle  had  been  a  feet  broader  and 
b  feet  longer,  it  would  have  been  c  square  feet  larger. 
But,  if  it  had  been  b  feet  wider  and  a  feet  longer,  it  would 
have  been  d  square  feet  larger.  Required  its  dimensions. 
Specialize  by  making  a  =  2,  b  =  3,  c  =  64,  and  d  =  68. 

17.  There  is  a  number  consisting  of  two  digits  whose 
sum  is  a,  and  if  b  be  subtracted  from  the  number,  the 
digits  will  change  places.  Required  the  number.  Special- 
ize by  putting  13  for  a  and  27  for  5. 

18.  The  wages  of  a  men  and  b  women  in  one  week 
amount  to  c  dollars,  and  b  men  receive  d  dollars  more  than 
e  women.  What  does  each  receive  per  week  ?  Put  5  for  a, 
7  for  b,  170  for  c,  80  for  d,  and  6  for  e. 

19.  Three  children,  taken  two  at  a  time,  weighed  a 
pounds,  b  pounds,  and  c  pounds.  What  was  the  weight 
of  each  ?    Fut  a  =  94,  b  =  76,  and  c  =  90. 

20.  A  purse  holds  a  crowns  and  b  guineas ;  c  crowns 
and  d  guineas  fill  ^Yes  of  it.  How  many  will  it  hold  of 
each  ?  Put  19  for  a,  6  for  b,  4  for  c,  and  5  for  d.  Enun- 
ciate the  special  problem  thus  formed. 


CHAPTER   IV. 
POWERS    AJ^B    ROOTS. 


Involution  of  Binomials. 

I.   Principle 
174.  We  may  learn  by  actual  multiplication  that : 

(a  +  Z»)3  =  «3  _^  3  «2  J  +  3  «  52  _|.  J3 

(a-\-bY  =  a""  +  4:aH -\-  Qa^h^ ^ 4.aW -\-¥ 
{a  -hY=ia''-4.aH^QaH^-^a¥-\-  5* 

By  a  careful  inspection  of  the  above  results  the  follow- 
ing laws  will  appear : 

The  Binomial  Theorem. 

Prin,  72, — 1.  The  number  of  terms  in  each  result  is 
one  greater  than  the  exponent  of  the  binomial, 

2.  When  the  binomial  is  the  sum  of  two  quantities,  all 
the  terms  of  the  power  are  positive  ;  when  the  difference  of 
two  quantities,  the  terms  are  alternately  positive  and  nega- 
tive. 

3.  The  first  letter  occurs  in  all  the  terms  but  the  last, 
and  the  second  letter  in  all  the  terms  but  the  first, 

Jf,  The  exponent  of  the  leading  letter  in  the  first  term  is 
the  same  as  the  exponent  of  the  binomial,  and  decreases  by 


162  ELEMENTARY  ALGEBRA. 

unity  in  each  succeeding  term.  The  exponent  of  the  second 
letter  is  one  in  the  second  ter?n,  and  increases  hy  unity  in 
each  succeeding  term. 

5.  The  coefficient  of  the  first  term  is  one  ;  of  the  second 
term,  the  exponent  of  the  Mnomial ;  and  that  of  each  suc- 
ceeding term  may  be  found  hy  multiplying  the  coefficient  of 
the  preceding  term  hy  the  exponent  of  the  leading  letter  in 
that  term^  and  dividing  the  product  hy  the  number  of  that 
term  from  the 


Note. — The  coefficients  after  the  middle  term  are  the  same,  in  an 
inverse  order,  as  those  before  it.  When  the  exponent  of  the  binomial 
is  odd,  there  are  two  middle  terms  with  like  coefficients. 

2.  Applications. 

EXERCISE    82. 

Expand  : 

1.  {c-\-dY  5.  {m-\-nf  9.  (x  —  yf 

2.  (a  -  df  6.  (m  -  nf  lo.  {c  +  zf 

3.  (x-\-yy  l.{c  —  xy  \i.{y  —  xf'' 

4.  (x  —  zf  8.  {x-\-zf  12.  {z-^yf^ 

13.  The  fourth  power  of  the  sum  of  two  quantities 
equals  what  ? 

Suggestion.— Since  {a -{-})f  =  a*  +  Aa^h  ■\- Qa^ b"^  +  4:ah^  +  h\  the 
fourth  power  of  the  sum  of  two  quantities  equals  the  fourth  power  of 
the  first  +  4  times  the  cube  of  the  first  into  the  second  +  6  times  the 
square  of  the  first  into  the  square  of  the  second,  etc. 

14.  The  4th  power  of  the  difference  of  two  quantities 
equals  what  ? 

15.  The  5th  power  of  the  sum  of  two  quantities  equals 
what? 

16.  The  5th  power  of  the  difference  of  two  quantities 
equals  what  ? 

17.  (:r+l)*=?  19.   {x^iy=?  21.   (1+;^)^=? 

18.  {x  -  1)*  =  ?  20.   {x  -  1)^  =  ?  22.   (1  -zf^'^ 


INVOLUTION  OF  BINOMIALS.  163 

Expand  (^x'-2fy. 

Solution :  Let  m  =  3  a:«  and  n  =  2  y\  then  (3  a:«  -  2  y^*  = 

(m  —  nf  =  m*  —  4:  w'  n  +  G  m^  n^  —  Amn^  +  n*  =  (3  a:*)*  — 

4  X  (3a;«)«  x  2y»  +  6  x  (3a:»)«  x  (23/3)2  _  4  x  3a:2  x  (22/')»  +  (2yY  = 

81a;«-216a:«y«  +  216a:*y«-9(>a;2  3/9  +  16  y«. 


Expand  : 

EXERCISE    83. 

1.  (a -{-2  by 

7.  (l-2a:2)5 

13.  (2a«  +  3a;3)« 

2.  (3a -25)* 

8.  (T'^fY 

14.   {-(x^-f)}^ 

3.  (3  a: +1)5 

9.  {a'  -  b'Y 

15.  (-2a:-3y)5 

^(-p)' 

■^  (-I)' 

...(i-i,)- 

••(^5)' 

/2  a      3JV 
''•[3 6- raj 

■'■(»''-•  ..y 

<-i)' 

-d-i")- 

...(..--■)• 

Involution  of  Polynomials. 
I.   Principles. 
175.  By  actual  multiplication  : 

1.  {a-{-b  -\-  cY  =  {a-{-b-{-  c)  (a-i-b-^  c)  = 

a^-^b^-^(^-]-2ab-\-2ac-^2bc. 

2.  {a-b-^cy  =  (a-b-\-c){a-b-j-c)  = 

a^-]-b^-i-(r-2ab-j-2ac-2bc, 

3.  (a  +  5  +  c4-^)*  =  a'  +  ^  +  c2  +  <^3  +  2a5  + 

2ac-\-2ad-\-2bc-\-2bd-i-2cd, 

4.  (a  -  5  +  c  -  J)2  =  a^  -j_  J2  _^  c2  +  ^2  _  2  « ^,  _|- 

2ac  — 2a^  — 2Jc  +  2Z'^  — 2c;<Z. 
Therefore, 

Prill,  73, — llie  square  of  a  polynomial  equals  the  sum 
of  the  squares  of  its  terms,  and  twice  the  product  of  each 
term  into  all  the  following  terms. 


164 


ELEMENTARY  ALGEBRA. 


176.  By  actual  multiplication  : 
,      1.  (a  +  ^'  +  c)3=(«  +  5  +  c)(«  +  ^  +  c)(«  +  J  +  c)  = 

Sc^a  +  ScH^t-^abc. 

2.  {a-h-\-cy={a-b-j-c){a-b-\-c)(a-b  +  c)  = 

a^-b^-\-c^-dan-^3a^c^3b^a-\-3b'-c  + 

Sc^a-3cH-6abc. 
Therefore, 

Frin,  74=. — The  cube  of  any  trinomial  equals  the  sum 
of  the  cubes  of  its  terms,  and  three  times  the  square  of  each 
term  into  all  the  other  terms,  and  six  times  the  product  of 
the  three  terms, 

2.  Applications. 

EXERCISE    84. 


Expand  : 

1.  (x^y-\-zY 

2.  {x  —  y  —  zf 

3.  {x-^y-^Xf 

4.  {a-b^  2)2 

5.  (a^-^ab-^b^Y 

6.  {%a^3b-cf 

8.  {;ix^hy-3c''f 
10.  {x  —  y^z  —  '^f 
12.  {m^-\-m^-\-m-^\y 


13.  {a^b-^Vf 

14.  {x  —  y  —  zf 

15.  {x^%^yf 

16.  (2  a; -3?/ +  5)3 

4^   '    6  7 

19.  {x-^'Zy-3zY 

20.  (a-  +  l+^y 

21.  (^+2-jy 

22.  (l  +  5a;  +  3a:2)3 
24.  {x?-Yx-  5)3 


ALGEBRAIC  EVOLUTION,  165 

Algebraic  Evolution. 

Definitions. 

177.  One  of  the  equal  factors  of  which  a  quantity  is 
composed  is  a  Root  of  the  quantity. 

Thus,  since  a^  =  a  X  a  X  a,  a  is  the  cube  root  of  a^. 

178.  The  number  of  equal  factors  into  which  a  quantity 
is  resolved  is  the  degree  of  the  root. 

179.  The  symbol  of  root  is  y^,  called  the  radical  sign, 

180.  The  degree  of  a  root  is  expressed  by  an  Index 
written  in  the  angle  of  the  radical  sign.  Thus,  the  fourth 
root  is  expressed  ^  ;  y'  =  */  is  the  square  root. 

181.  The  process  of  obtaining  a  root  of  a  quantity  is 
evolution. 

Principles. 

182.  Since  (^2)3  =  ^2x3  [p.  29]  =  ««,   Va^  =  «''■"  ^  =  a^ 

Therefore, 

Prin,  75. — Dividing  the  exponent  of  any  factor  hy  the 
index  of  a  root  takes  that  root  of  the  factor. 

SIGHT      EXERCI  SE. 

Name  at  sight : 

1.  \/^  4.   V^  7.  'V?«  10.  'v^ 

2.  V^  5.   V^  8.   Vc^  11.  V^ 

3.  Vc^  6.  'Vf'  9.  Vn^  12.  'V^ 

183.  Since  (aH^d'f  =  {a'Y  x  (^)*  X  (^)*  [P.  30]  = 
aH''c'\   Van'^c^^=  V^ X  v^  X  Vc^  =  a^Pd', 

Therefore, 

JPrin.  76, — Taking  any  root  of  every  factor  of  a  quan- 
tity takes  that  root  of  the  quantity. 


166  ELEMENTARY  ALGEBRA, 

SIGHT      EXERCISE. 


Name  at  sight : 


1. 

^/aH"- 

2. 
3. 

6.   V^V^  9-  V4  X  9  X  16 


6.  Va^2^24^i8  ^Q    VS  X  27  X  64 


7.   VS^V^  11-  -Vl6a*^>8 


4.  Va^«^2o  8.  Va}H^z^  12.  V«^«(a  +  ^>)25 

184.  Any  even  power  of  a  positive  or  a  negative  quantity 
is  positive  [P.  27].     Therefore, 

Prin.  77 » — Any  even  root  of  a  positive  quantity  may 
ie  either  positive  or  negative. 


Illustration  :  a/+  64  =  ±  8,  since  (±8)^  =  4-  64. 

185.  Any  odd  power  of  a  quantity  has  the  same  sign 
as  the  quantity  [P.  28].     Therefore, 

Prin,  78, — Any  odd  root  of  a  quantity  has  the  same 
sign  as  the  quantity. 

lUustration  :  V+ 27  =  +  3,  since  (+  3)^  =  +  27. 
V-27  =  -  3,  since  (-  3)^  =  -  27. 

186.  Since  no  even  power  is  negative  [P.  27], 

Prin,  79, — An  even  root  of  a  negative  quantity  is  im- 


Illustration  :  v  —  16  is  neither  ±  4,  since  (±  4)^  =  16. 

Note. — The  indicated  even  root  of  a  negative  quantity,  as  V—  16, 
is  called  an  imaginary  quantity. 

SIGHTEXERCISE. 

Name  at  sight,  giving  the  proper  signs  : 


1.  Vl6 

4.  V81 

5.  V-X^ 

6.  V-32 

7.   V«* 

10.  VaH'^ 

2.  V8 

3.  V-27 

8.  V-x^y^ 

9.  Vx^^ 

11.  V-4. 

12.  V-16 

ALGEBRAIC  EVOLUTION, 


167 


187.  Since  raising  both  terms  of  a  fraction  to  any  power 
raises  the  fraction  to  that  power  [P.  68], 

rrin.  80, — Extracting  any  root  of  both  terms  of  a  frac- 
tion extracts  that  root  of  the  fraction. 

niustration. — 


±-^=±^,  since 


\     3/        ^  (3f      ^  9 


SIGHT      EXERCI  SE. 
3    /  X 


3a 


10. 


11. 


•y-3^ 

3 /T     T 

V,27^64 
y       8  ^  27 


Problem  1.    To  find  a  root  of  a  numerical  quantity  by 
factoring. 


niustration. — 

Find  the  cube  root  of  1728. 

Solution:  Since  the  cube  root  of 
a  number  is  one  of  the  three  equal 
factors  of  the  number,  we  resolve 
1728  into  its  prime  factors,  and  take 
one  of  every  three  equal  ones,  and 
find  their  product. 

Note. — To  find  the  square  root, 
take  one  of  every  two  equal  factors ; 
to  find  the  fourth  root,  one  of  every 
four  equal  factors,  etc. 


Form. 

2 

1728 

2 

864 

*2 

432 

2 

216 

2 

108 

*2 

54 

3  1      27 

3 

9 

*3 

Vl728  =  2X2X3  =  12 


168 


ELEMENTARY  ALGEBRA, 


EXERCISE    8B. 

Find  the  value  of  : 

1.  V324  4.  V512 

2.  Vl296  5.  V3375 

3.  ^2304  6.  V5832 


7.  V4096 

8.  V20736 


9.  V248832 


Problem  2.    To  find  a  root  of  a  xnonoxnial. 

Ulustration. — 

Find  the  5th  root  of  ^o™* 

Solution:    Since  taking  a 
root  of  every  factor  of  a  quantity  takes  the  root  of  the  quantity  [P.  76], 

^  X  V^  X  V^  X  v^.   y^^33 


=  -2  [P.  78];    V^=ia^,    V^=Z»,  and   V^=6'*  [P.  75]; 
hence  the  result  is  —  2  «^  ^  C*.      Therefore, 

Mule. — Tahe  the  required  root  of  the  numerical  co- 
efficient and  divide  tJie  exponent  of  each  literal  factor  hy  the 
index  of  the  root. 


EXERCISE    86. 


Find  the  value  of 


1.  Vc^YJ 


8.  VlMo^^V^ 


2.  V4.anu^^ 


9.   V- 729  (6^  +  0:)^ 


3.   Vx^y'z'^ 


4k.  V8 


10.  V256(«-:r)« 


twn'^ 


5.  V-27a:«y2 

6.  wmFvFp^ 

7.  V-^^a^H^'^c^ 


11.  V-(«  +  Zi)5ci« 

12.  VlOOOOa:»(a;  +  ^)i» 

13.  V-  243  {m  +  w)^« 

14.  V64  (:^  -  «/2)i2 


15.  V^^y^x^^~yY,  when  a;  =  6  and  ?/  =  3 


16.   Vie  flS  ^,12  (^2  _  j,2Y^  ^hen  a  =  5  and  J  =  3 


ALGEBRAIC  EVOLUTION.  169 


17.  V(ar  -  b^f  -f-  V{a  +  bf  when  a  =  3  and  J  =  2 

18.  l/(a  +  a:)«  +  V(a  --  a:)^  -|-  VC^^  -  3^)\  when  a  =  4 

and  a;  =  2 

19.  /Va«1^+  f'Va^^-  Va^,  when  a  =  -  3 

and  J  =  5 
Find  the  value  of  : 

y  16'  r  9'  T  16'  y  25'  y  si 

^/Z      3  /_i  V  ^      3/       8a:«y»"      3  /a;«  (<^  -f  a;)3' 

y27'  y  8^^^'  y  27«=^i^2'  y  ^9^18 

^'*-  y  («_^)4'  y    (a;+i/)»'  y  (a-^)«'  y    z%a>^ 

y  27'  y  3 ^  9'  y  128'  y  («-a:r'  y  m 


20. 


21. 


22. 


Problem  3.    To  extract  the  square  root  of  a  polynomiaL 
I.   Method. 

188.  Since  the  square  of  a  polynomial  is  the  sum  of  the 
squares  of  its  terms  and  twice  the  product  of  each  term 
into  all  the  following  terms  [P.  73],  the  square  root  of  a 
polynomial  that  is  a  perfect  square  may  be  obtained  by 
inspection,  if  no  terms  of  the  power  have  disappeared  by 
collection. 

Illustration. — 


V(a»-h2a^4-^  +  ^gg  +  ^^g  +  g^)  = 
'v/(a*  +  J*  +  c24-2ad  +  2flrc-f~2T(c)  =  a  +  J  +  c,  since 
{a-{-h-\-cY  =  a--\-¥-\-c^^2ab-\-'ilac-\-%bc, 

{    UNIVEP 


170 


ELEMENTARY  ALGEBRA, 


EXERCISE    87. 

Extract  the  square  root  of  : 
\.  a^-\-¥^^al)  3.  a;2  +  16  +  8:2; 

2.  x^-{-y^  —  2xy  4.  :r2  —  6  :r  +  9 

5.  x^-\-y^-\-z^-\-%xy-\-2xz-\-2yz 

6.  x^-\-y^-^z^  —  %xy  —  'ilxz-\-%yz 

7.  a^-\-^h^-\-^c^-\-^al)^^ac-\-l'^hc 

8.  4.x^-^y^-{-^  —  4.xy^l%x-'^y 

9.  x^-^y^-i-2x^y-^2x^  +  2y-}-l 
10.  9x^-24:xy  +  16y^ 

12.  4:X^-{-^y^-i-9-^2x^y-{-12x^-^3y 

189.  When  the  law  of  development  does  not  appear  by 
inspection,  the  following  method  must  be  resorted  to. 
Illustration. — Extract  the  square  root  of 

x^-\-f-\-3x^y^-2x^y-2xy\ 

7n-\-n 


{m  +  nY  =  m^  -\-  {2  m -\- n)  n 


m-^  71 


2a^y-\-3x^y^  —  2xy^-\-y^\  x^  —  xy -\-y^ 


x^ 


%x^  —  xy 


2x?y  -\-Zx?  y^ 
2x^y  -\-    x^y^ 


2  o;^  —  2  :c  y  +  y^ 


2xry^ 
2a^y^ 


2xy^-^f 
2xy^-\-y^ 


Solution  :  Having  arranged  the  terms  according  to  the  descending 
powers  of  x  assumed  as  the  leading  letter,  we  will  proceed  to  take 
out  of  the  polynomial  the  square  of  the  first  two  terms  of  the  root. 
For  this  purpose  we  let  m  +  w  represent  the  first  two  terms  of  the 
root.  Now  {m  +  nf  =  m^  +  2mn  +  n^,  or  m"^  +  {2m  +  n)n.  m'  ob- 
viously equals  a:*,  or  w  =  x^.     Subtracting  ic*  from  the  polynomial 


ALGEBRAIC  EVOLUTION.  171 

and  bringing'  down  the  next  two  terms,  we  have  —  2.r'y  +  3a;'y'. 
This  remainder  consists  mainly  of  (2  w  +  n)  n ;  hence  if  we  use  3  w, 
or  2x',  as  a  trial  divisor^  we  will  obtain  the  value  of  w,  which  is 
— -  2  x*  y -*- 2  a;',  or  —xy;  adding  this  value  of  n  to  that  of  2  m,  we 
have  2x^—xy,  the  complete  divisor ;  multiplying  the  value  of  2m  +  n, 
or  x^  —  X y,  by  the  value  of  n,  or  —xy,  we  have  —  2 a:* y  +  a;* y*. 
Subtracting  this  product  from  —2a^y  +  ^x^y^  and  bringing  down 
the  remaming  terms,  we  have  2x^y^  —  2xy^  +  y*. 

We  now  let  m  represent  x^  —  xy  and  n  the  next  term  of  the  root, 
and  proceed  as  before  to  take  out  of  the  polynomial  the  square  of 
m  +  n,  or  w*  +  (2  m  +  n)  n.  m^  or  (x^  —  x  yf  has  already  been  re- 
moved, hence  the  remainder  2x^y  —  2xy'^  -v  i^  is  composeti  of 
(2 m  +  n) n.  Using  2 w,  or  2x^  —  2xy,  as  a  trial  divisor,  we  obtain 
y'  for  n ;  adding  this  to  the  value  of  2 m,  or  2x^  —  xy,  we  have 
2x'^  —  2xy-\-y^  for  the  complete  divisor.  Multiplying  the  complete 
divisor  by  y'  and  subtracting,  nothing  remains.  Therefore  the  given 
polynomial  is  the  square  of  x?  —  xy  •\-y^,  or  x^  —  xy  -^  y^  is  the 
square  root  required. 

From  an  inspection  of  the  above  solution  the  following 
rule  will  appear  : 

1.  Arrange  the  terms  of  the  polynomial  according  to  the 
ascending  or  descending  powers  of  some  letter  assumed  as  a 
leading  letter, 

2.  Take  the  square  root  of  the  first  term  of  the  poly- 
nomial for  the  first  term  of  the  root.  Subtract  the  square 
of  this  term  of  the  root  from  the  polynomial. 

3.  Double  the  root  found  for  a  trial  divisor.  Divide 
the  first  term  of  the  remainder  by  the  trial  divisor  for  the 
next  term  of  the  root. 

4.  Add  the  last  term  of  the  root  found  to  the  trial  divisor 
for  the  complete  divisor.  Multiply  the  complete  divisor  by 
the  last  term  of  the  root  found,  and  subtract  the  product 
from  the  remainder,  and  bring  down  such  terms  as  are 
needed. 

5.  If  the  root  has  more  than  two  terms,  double  the  root 
already  found  for  a  new  trial  divisor,  and  proceed  as  be- 
fore to  obtain  the  next  term  of  the  root  and  the  complete 
divisor.  Continue  this  process  until  all  the  terms  of  tlie 
polynomial  have  been  used. 


172  ELEMENTARY  ALGEBRA. 

^ote. — In  the  formula  m^  +  (3  m  +  w)  n,  m  represents  the  root  as 
far  as  found,  2  m  the  trial  divisor,  n  the  next  term  of  the  root  and 
also  the  correction,  and  2  m  +  n  the  complete  divisor. 

EXERCISE    88. 

Extract  the  square  root  of  : 

1.  x^-{-2a^-{-3x^-]-2x-^l 

2.  x^-4:X^-^ex^-4:X-^l 

3.  x^-^4.x^  +  10x^-\-12x-}-9 

4.  x^-6x^y-\-lda^y^-12xy^-\-4.f 

5.  x^-4:X^-\-10x^-12x^-i-9x^ 

6.  4:X^  —  20x^y  +  S7x^y^-dOxy''^df 

•'■^'  +  ^  +  |^^  +  i^  +  ^ 

8.^^  +  4:^:^  +  6  +  ^  +  1 

9.  x'^ -]-  2x^  —  x^ -]- 3 x^  —  2 X -\-l 

Problem  4.    To  extract  tlie  square  root  of  numerical 
quantities. 

I.   Definition  and   Principle. 

190.  Every  numerical  quantity  of  two  or  more  figures 
may  be  considered  a  polynomial.     Thus, 

123456  =  12  ten-thousands  +  34  hundreds  +  56  units. 

191.  The  square  of  a  unit  is  a  unit,  the  square  of  a  ten 
is  a  hundred,  the  square  of  a  hundred  is  a  ten-thousand, 
the  square  of  a  thousand  is  a  million,  etc. ;  hence,  the 
square  denominations"  in  order  are  the  unit,  the  hundred, 
the  ten-thousand,  the  million,  etc.     Therefore, 

Prin,  81. — i/"  a  number  he  pointed  off  into  terms  of  two 
figures  each,  beginning  at  the  units,  the  unit  of  each  term 
will  J)e  a  perfect  square. 


ALGEBRAIC  EVOLUTION. 


173 


2.  Examples. 
UlustratioiL — Extract  the  square  root  of  105625. 


Fornit 

1  0'5  6'2  5 
9 


325 


62 


156 
124 


645 


3225 
3225 


Solution:  We  point  the  number 
off  into  terras  of  two  figures  each  to 
keep  the  unit  of  each  terra  a  perfect 
square  [P.  81].  105635  equals  10  ten- 
thousands  +  56  hundreds  +  25  units. 

The  square  root  of  10  ten-thou- 
sands is  3  hundreds,  the  first  term  of 
the  root.  Squaring  3  hundreds,  we 
have  9  ten-thousands;  subtracting  9 
ten-thousands  from  10  ten-thousands 
and  bringing  down   the  next  term, 

we  have  156  hundreds.  Doubling  the  root  already  found  for  a  trial 
divisor,  we  have  6  hundreds ;  dividing  15  thousands  by  6  hundreds, 
we  have  2  tens  for  the  next  terra  of  the  root ;  adding  2  tens  to  the 
trial  divisor,  we  have  62  tens  for  the  complete  divisor;  multiplying 
62  tens  by  2  tens,  we  have  124  hundreds;  subtracting  124  hundreds 
from  156  hundreds  and  bringing  down  the  next  term,  we  have  3225 
units.  Doubling  32  tens,  we  have  64  tens  for  a  new  trial  divisor; 
dividing  322  tens  by  64  tens,  we  have  5  units  for  the  next  term  of 
the  root ;  adding  5  units  to  64  tens,  we  have  645  units  for  the  cora- 
plete  divisor;  multiplying  645  units  by  5,  we  have  3225  units;  sub- 
tracting this  product  from  3225,  nothing  remains.  Therefore  the 
square  root  of  105625  is  325. 

Note. — The  square  root  may  also  be  obtained  by  means  of  the 
formula  {m  +  nf  =  m'  +  (2  m  +  n)  ;i,  as  in  example  (Art.  189). 


EXERCISE    89. 

Extract  the  square  root  of  : 

1.  289  6.  2704 

2.  676  6.  4761 

3.  1225         7.  5041 

4.  1849         8.  7056 

13.  What  is  the  value  of  : 
(•1)2?  (-01)2?  (-001)*? 

V-01?      V-0001  ?      V -000001  ?      V -00000001  ? 


9.  16129 

10.  60025 

11.  104976 

12.  166464 


(-0001) 


2? 


174 


ELEMENTARY  ALGEBRA. 


192.  The  square  decimal  units  below  one  are  the  hun- 
dredth, the  ten-thousandth,  the  millionth,  the  hundred- 
millionth,  etc.     Therefore, 

193.  If  a  decimal  contain,  or  be  made  to  contain  by 
annexing  ciphers,  an  even  number  of  figures,  its  unit  will 
be  a  perfect  square. 

Illustration. — 

•24  =  24  X  -01  -536420  =  536420  X  '000001 

•3645  =  3645  X  '0001        5-000000  =  5000000  X  '000001 

14.  Extract  the  square  root  of  5  to  thousandths. 

Solution :  Since  the  square  P 

root  is  to  be  expressed  in  >      i      , 

thousandths,     the     number  "        5*000000  |2-2364- 

must    be   reduced    to    mill-  4 

ionths.  5  =  5,000,000  mill- 
ionths.  The  square  root  of 
5,000,000  is  2236  +,  and  the 
square  root  of  a  millionth  is 
a  thousandth.  Therefore  the 
square  root  of  5,000,000  mill- 
ionths  is  2236  +  thousandths, 
or  2-236  +  . 

Extract  the  square  root  of  : 

15.  -0049  19.  -00000016  23.  108-5764 

16.  -0625  20.  -104976  24.  1024-6401 

17.  -000144  21.  33-8724  25.  99-980001 

18.  882-09  22.  11-4244  26.  8010-25 

27.  Find  the  square  root  of  10,  11,  12,  and  13  to  within 
one  ten-thousandth. 

7     .  /T 


to  4 


28.  Find  the  value  of  V2, 
decimal  places. 

29.  Find  the  value  of   a/40,   V4i,   a/42,  and  a/43  to 
within  one  thousandth. 


42 

100 

84 

44: 

5 

1600 
1329 

446( 

3 

27100 
26796 

ALGEBRAIC  EVOLUTION. 


175 


Problem  5.     To  extract  the  cube  root  of  a  polynomiaL 
Illustration. — Extract  the  cube  root  of 

Fornu  ^'^  +  ^i 

(m  +  n)'  =  m»  +  (3m»+3mn  +  n«)/»  w+n 

x^->fxy-\-y^ 


T.  D.     =3x4 

1st  Cor.  =          Z7?y 

2d  Cor.  =                      xhf 

CD.      =3x4  +  3a;»i^+a;V 

3a;»y  +  6x^^2  +  7u;V 

T.D.     =3x*  +  6x»i/  +  3xV 
1st  Cor.  =                      3a;V- 
2d  Cor.  = 

f3xy» 

3x*2/*  +  «-^/  +  6a:V+3a:3/*+3^ 

C.  D.      =  3a;4+6a:»y+6xV  +  3a:!/3  +  y* 

3  a;4y' +  6  a:  V  +  6  ;'^y  +  3  X2/6  +  3/« 

Solation  :  Having  arranged  the  terms  according  to  the  descending 
powers  of  x,  assumed  as  the  leading  letter,  we  proceed  to  take  out  of 
the  polynomial  the  cube  of  the  first  two  terms  of  the  root.  For  this 
purpose  we  let  wi  +  w  represent  these  terms.  Now,  (m  +  ixf  =  m^  + 
3  m'  7t  +  3  w  n^  +  w',  or  m^  +  (3  m^  +  3  m  w  +  w')  n.  m^  obviously  equals 
JK*,  or  m  equals  a;'.  Subtracting  x^  from  the  polynomial  and  bringing 
down  three  terms,  we  have  for  the  first  remainder,  3a:^y  +  6a:*2/*  + 
7  x*  y*.  This  remainder  consists  mainly  of  (3  w*  +  3  m  n  +  w*)  n ;  hence, 
if  we  use  3m^  or  3  a:*,  for  a  trial  divisor,  we  will  obtain  the  value 
of  n,  which  is  3  aH*  y  -s-  3  a:*,  or  xy.  Substituting  the  values  of  m  and 
n  in  Zmn  and  w*,  we  have  3 a;* y  for  the  first  and  x^ y^  for  the  second 
correction ;  adding  the  two  corrections  to  the  trial  divisor,  we  have 
3a:*  +  3a:^y  +  a:*2/'  for  3m*  +  3m7i  +  n*,  the  complete  divisor ;  mul- 
tiplying the  complete  divisor  by  xy,  the  value  of  n,  we  have  3a:'y  + 
3a:*y'  +  a:^2/' ;  subtracting  this  from  the  first  remainder,  and  bringing 
down  the  remaining  terms,  we  have  3a:*y^  +  6a:'i/'  +  6a:*y*  +  3a:2/'+y*. 

We  now  let  m  stand  for  x*  +  xy,  the  root  already  found,  and  n 
for  the  next  term  of  the  root,  and  proceed  as  before  to  take  from  the 
polynomial  the  cube  of  m  +  n,  or  m*  +  (3  m'  +  3  m  n  +  w*)  w.  m^,  or 
(a:*  +  X  y)\  has  already  been  subtracted ;  hence,  the  remainder  consists 
of  (3w^  +  3mw  +  n'^)n.  Using  as  before  3  w^  or  3a:*  +  Qx^y  +  3x*y', 
as  a  trial  divisor,  wo  obtain  y^  for  ?» ;  substituting  the  values  of  m 
and  n  in  3 win  and  w',  we  have  3 a;' y'  +  3 a; y*  for  the  first  and  y* 
for  the  second  correctim,  and  3a:*  +  6a:*y  +  6a:*y*  +  3a:y*  +  2/*  for 


176  ELEMENTARY  ALGEBRA. 

the  complete  divisor.  Multiplying  the  complete  divisor  by  2/^  and 
subtracting  the  product  from  the  last  remainder,  nothing  remains. 
Therefore,  x^  +  xy  +  y^  is  the  cube  root  required. 

Note. — In  the  formula  m^  +  (Sm^  +  Smn  +  n*)n,  m  staTids  for  the 
root  already  found,  3 m^  for  the  trial  divisor,  Smn  for  the  first  cor- 
rection, n^  for  the  second  correction,  d  m^  ■{■  3  m  n  +  n^  for  the  com- 
plete divisor,  and  n  for  the  next  term  of  the  root. 

From  an  inspection  of  the  above  solution  the  following 
rule  will  appear : 

1.  Arrange  the  terms  of  the  polynomial  according  to  the 
ascending  or  descending  powers  of  some  letter  assumed  as  a 
leading  letter, 

2.  Take  the  cube  root  of  the  first  term  of  the  polynomial 
for  the  first  term  of  the  root.  Subtract  the  cube  of  this 
term  of  the  root  from  the  polynomial. 

S.  Take  three  times  the  square  of  the  root  already  found 
for  a  trial  divisor.  Divide  the  first  term  of  the  remainder 
by  the  trial  divisor  for  the  next  term  of  the  root. 

Jj..  Add  to  the  trial  divisor  three  times  the  last  term  of 
the  root  found  multiplied  by  the  preceding  part  of  the  root, 
and  the  square  of  the  last  term  found,  for  a  complete  di- 
visor. Multiply  the  complete  divisor  by  the  last  term,  of 
the  root  found,  and  subtract  the  result  from  the  remainder, 
bringing  down  only  such  terms  as  are  needed. 

5.  If  the  root  has  more  than  two  terms,  take  three  times 
the  square  of  the  root  already  found  for  a  new  trial  divisor, 
and  proceed  as  before  to  obtain  the  next  term  of  the  root, 
the  new  corrections,  and  the  new  complete  divisor.  Con- 
tinue the  process  until  all  the  terms  have  been  used. 

EXERCISE    90- 

Extract  the  cube  root  of  : 

1.  a:^  +  3a^  +  3a;  +  l  4.  8a;«  + 36ir*  + 54.^'2  + 27 

2.  a^-^a''b^ZaJy'-¥  b.  x^ -\-^x^  -  bx^ -\-Zx-l 

3.  ic3-j-12a:2  4-48a;-f64  6.  i/^  -  3^  +  5  «/3  -  3y  -  1 

7.  a^a^  —  ^a'^bx^-^-daWx-b^ 

8.  Sa^x^-dQaHx^y  +  h4.al^xy^-21b^f 


ALGEBRAIC  EVOLUTION. 


177 


9.  a:^  -  6  ar^  +  21  r^:*  -  44  2^3  _^  53  ^  _  54  2.  _j_  27 

12 


10.   Q^  +  ^X^--\-\ 


11.  a^o^  —  Qax-{- 


8 


ax      d^7? 


12.  ^«  +  3a:*  +  6a:^+7  +  |  +  |  +  ^ 


13.  a^-\-¥^(?^Zd^l^Za^c-\-ZaW-^Z}rc^ 


Problem  6.  To  extract  the  cube  root  of  niunerical  quantities. 

1.  Principle. 

194.  The  cube  of  a  unit  is  a  unit,  the  cube  of  a  ten  is 
a  thousand,  the  cube  of  a  hundred  is  a  million,  the  cube 
of  a  thousand  is  a  trillion,  etc. ;  hence,  the  cubic  denom- 
inations in  order  are  the  unit,  the  thousand,  the  million, 
the  trillion,  etc.     Therefore, 

Prin,  82, — 7/"  a  number  he  pointed  off  into  terms  of 
three  figures  each,  beginning  at  the  units,  the  unit  of  each 
term  will  be  a  perfect  cube. 

2.  Examples, 
niustration.— Extract  the  cube  root  of  16387064. 

ForiQf 

(m -\-nf  =  m^-\-{Zm^  +  ^mn-\- n^)  n 

16'387'064 

^3=      _8 

3^2    =3x(2..)2    =12.  .  .  .       8387 
Zmn  =3X2..X5.=     30  .  . 
n-     =  (5.)2=       25. 


in-¥n 
2   5     4 


3  7^2-1-3  w.  71-1- ;r 


=  1525 


7625 


37/12    =3X(25.)2    =     1875 

37/i7i  =3x25.X4  =  300. 

w*     =  42=  16 


762064 


37^^4-3  m  n-\-n^ 


=     100516      762064 


178  ELEMENTARY  ALGEBRA. 

Solution  :  We  point  off  the  number  into  terras  of  three  figures  each 
to  make  the  unit  of  each  term  a  perfect  cube  [P.  82J.  We  find  thus 
that  16,387,064  =  16  million,  +  387  thousand,  +  64  units.  The  cube 
root  of  16  million  is  2  hundred  +  .  Cubing  2  hundred,  we  have  8 
million ;  subtracting  8  million  from  16  million  and  bringing  down  the 
next  term,  we  have  8387  thousand.  Taking  3  times  the  square  of  the 
root  already  found  (3  m^)  for  a  trial  divisor,  we  have  12  ten-thousands ; 
dividing  83  hundred-thousands  by  12  ten-thousands,  we  have  5  tens  (7?), 
the  next  term  of  the  root ;  taking  3  times  the  root  previously  found 
(3  m)  and  multiplying  it  by  the  last  term  found  (n),  we  have  30  thou- 
sand for  the  first  correction,  and  squaring  the  last  term  of  the  root  {n% 
we  have  25  hundred  for  the  second  correction ;  adding  the  trial  divisor 
and  the  two  corrections,  we  have  1525  hundred  for  the  complete  divisor 
(3  m^  +  3  m  w  +  71^) ;  multiplying  the  complete  divisor  by  the  last  term 
of  the  root  {n\  we  have  7625  hundred ;  subtracting  7625  hundred  from 
8387  hundred,  and  bringing  down  the  next  term,  we  have  762064  units. 

Taking  3  times  the  square  of  25  tens  (the  new  value  of  m),  we  have 
1875  hundred  (the  new  trial  divisor) ;  dividing  7620  hundred  by  1875 
hundred,  we  have  4,  the  next  term  of  the  root  {n) ;  finding  as  before 
the  values  of  3  m  w  and  w^,  we  have  for  the  two  corrections  300  tens 
and  16  units,  and  for  the  complete  divisor  (3  m^  +  3  m  /i  +  n^)  190516 ; 
multiplying  by  4,  or  n,  we  have  762064,  which  subtracted  from  762064 
leaves  nothing.    Therefore  the  cube  root  of  16,387,064  is  254. 

Abbreviated    Rule. 

1.  Point  off  the  nwnber  into  terms  of  three  figures  each, 

2,  The  cube  root  of  the  first  term  gives  the  first  figure 
of  the  root. 

S.  Three  times  the  square  of  the  root  already  found 
always  gives  the  trial  divisor. 

J/..  The  remainder,  exclusive  of  the  two  right-hand  fig- 
ures, divided  hy  the  trial  divisor,  gives  the  next  figure  of 
the  root. 

5.  Three  times  the  root  previously  found  multiplied  hy 
the  last  figure  found  gives  the  first  correction,  and  the 
square  of  the  last  figure  found,  the  second  correction. 

6.  The  right-hand  figure  of  the  first  correction  is  placed 
one  order  to  the  right  of  the  trial  divisor,  and  that  of  the 
second  correction  one  order  to  the  right  of  the  first  correction. 

7.  The  sum  of  the  trial  divisor  and  the  two  corrections 
gives  the  complete  divisor. 


ALGEBRAIC  EVOLUTION. 


179 


EXERCISE    91. 

Extract  the  cube  root  of  : 

1.  2744         5.  373248  9.  1815848 

2.  19683        6.  592704  10.  10941048 

3.  42875        7.  681472  11.  28372625 

4.  300763       8.  941192  12.  74088000 

13.  What  is  the  value  of  : 

(•1)3?        (-01)3?    '     (-001)3? 

V-001  ?  V -000001  ?  V -000000001? 

195.  The  cubic  decimal  units  below  one  are  the  thou- 
sandth, the  millionth,  the  billionth,  the  trillionth,  etc. 
Therefore, 

196.  If  a  decimal  contain,  or  be  made  to  contain  by 
annexing  ciphers,  a  whole  number  of  times  three  figures, 
its  unit  will  be  a  perfect  cube. 

niustration  :    -325  =  325  X  '001  ; 

4-25  =  4-250000  =  4250000  X  '000001 

14.  Extract  the  cube  root  of  3*25  to  hundredths. 

Form. 

3-'2  5  0'0  0  0[lj47 
1 

3  X  P  =  3 
3X1X4  =12 
42=     16 


436 


2250 


1744 


3  X  142  =  5  88 
3X14X7  =    294 
72=  49 


61789 


506000 


432523 


Solution :  Since  the  cube  root  is  to  be  expressed  in  hundredths,  the 
number  must  be  reduced  to  millionths.  3*25  =  3,250,000  millionths. 
The  cube  root  of  3,250,000  is  147 +  ,  and  the  cube  root  of  1  millionth 
is  1  hundredth.  Therefore  the  cube  root  of  3,250,000  millionths  is 
147+  hundredths,  or  1-47+. 


180  ELEMENTARY  ALGEBRA, 

Extract  the  cube  root  of  : 

15.  -008  18. '-015625  21.  39-304 

16.  -001728  19.  -029791  22.  13-824 

17.  -000027  20.   -125000  23.  8I-37V27 

24.  Find  the  value  to  thousandths  of  V^,    A/  — ,   V-27 
Problem  7.     To  extract  higher  roots  of  quantities. 


197.  Since  {ay=a\  Va^=Wa^;  also,  smce  (aY=a\ 


Va^=VV^;  again,  since  (a^)'^  =  a^,   X/c^=Vl/c^, 

Therefore, 

To  extract  the  fourth  root  of  a  quantity,  extract  the 
square  root  of  the  square  root ;  to  extract  the  sixth  root, 
extract  the  cube  root  of  the  square  root;  and  to  extract 
the  ninth  root,  extract  the  cuhe  root  of  the  cube  root, 

EXERCISE    92. 

1.  Extract  the  fourth  root  of 

2^  +  4ic«/ +  6  a:V  +  4^^2/^  +  2/^ 

2.  Extract  the  sixth  root  of 

a;6  -f  6  ^5y  +  15  2;*  1/2  _|_  20 a:3^3  _|_  15  ^2^  _|.  g  ^^5  _}_  ^e 

3.  Find  the  value  of  V256,   V729,   Vl953125,   -V-0016 


Factoring  with  the  aid  of  Evolution. 

198.  If  a  polynomial  is  the  difference  of  the  squares, 
the  difference  of  the  cubes,  or  the  sum  of  the  cubes  of  two 
quantities,  it  may  be  factored  by  P.  39,  44,  43. 

niustrations.— 1.  Factor  x'^ -\-%x^  ^ha? -{-^x-\-^. 

Solution :  By  extracting  the  square  root  of  the  given  polynomial 
we  find  that  it  lacks  1  of  being  the  square  of  a:;^  +  a;  +  2, 

.-.    a;*  +  2a;3  +  52;2  +  4a:  +  3  =  (a;2+a;  +  2)2-l  = 

(a;2  +  cc  +  2  +  1) (a;2  +  a;  +  2  -  1)  [P.  39]  =  {x^  +  x  +  ^){x^  +  x  +  1). 


FACTORING  WITH  THE  AID  OF  EVOLUTION.     181 

2.  Factor  oi? -[-Qx^ +l^x-\-l. 

Solution  :  By  extracting  the  cube  root  of  a:^  +  6  a;*  +  12  a;  +  7,  we 
find  that  it  lacks  1  of  being  the  cube  of  a;  +  2, 
.-.    a? +  Qx^ +  \2x-k-l  =  ix  +  2f-l; 

,'.    a:^  +  6a;«  +  12a;  +  7  is  divisible  by  a;  +  2  —  1  or  a;  +  1  [P.  44] ; 
.-.    a;3  + 6a;«  + 12a;  + 7  =  (a;  + l)(a;»  +  5a;  + 7). 

3.  Factor  x''-{-3ax^-{-3a^a^-\-9a\ 

Solution :  By  extracting  the  cube  root,  or  by  inspection,  we  see 
that  a;*  +  3  a  a;*  +  3  a*  a;*  +  9  a'  is  8  a'  more  than  the  cube  of  a;'  +  a, 
.-.    a:«  +  3aa:*  +  3a*a:»  +  9a»  =  (a;«  +  a)»  +  8a«; 
.  • .    a;*  +  3  a  a;*  +  3  a*  a;*  +  9  a'  is  divisible  by  a;*  +  a  +  2  a, 

or  a;«  +  3a  [P.  43]; 
.-.    a:«  +  3  aa;*  +  3  a^a;^  +  9a«  =  (a;2  +  3  a)  (a;4  +  3  a»). 

EXERCISE    93. 

Factor  : 

1.  3^-j-a'^a:^-{-a^  4.  25  a^  -  9  r^  y^ -{- 16  y* 

2.  a^^ex-}-6  5.  4jo*-37yg2  +  9^ 

3.  4.a^  +  12xy-^6y^  6.  64a*  +  128^2^  + 81^* 

7.  a^-^2:x^-\-33^-{-2X'-3 

8.  4:a^-{-20x^y-{-29x^y^  +  10xf-3y* 

9.  8a:3_j_60a:2  4-l50a;  +  61 

10.  27a^-54:a^y-}-36xy^-7y^ 

11.  a;«  +  6a^«/8  +  12a:2^_19^6 

12.  8a^-12a'b^-\-6aH'-9b^ 

13.  a^-]-303^-\-300x^S75 

14.  8  a»  -  12  a«  +  6  a^  +  7 

15.  fl^  +  3a«^>3_|_3^3j6_|_9j9 

16.  a^2_3a8J4_p3^4  58_28J12 

17.  4a2+12aJ  +  8^>2_^16ac  +  225c  +  15c3 

18.  27a3-135a2&  +  225a^-61J^ 

19.  a»a;3_^3^j2^y^3^2j4a.^2_7j6^ 


CHAPTER  V. 
QUADRATIC    EQUATIOJTS. 


Quadratic  Equations  of  One  Unknown  Quantity. 
Pure  Quadratics. 

I.   Definitions  and   Principles. 

199.  A  quadratic  equation  containing  only  the  second 
power  of  an  unknown  quantity  is  a  pure  or  incomplete 

quadratic  equation  ;  as,  3  a:^  =  8,  or  —  +  — -  =  - . 

ODD 

200.  Every  equation  containing  fractional  terms  may 
be  cleared  of  fractions  [P.  71].  All  the  unknown  terms 
in  the  second  member  may  be  transposed  to  the  first,  and 
all  the  known  terms  in  the  first  member  to  the  second 
[P.  70].  All  the  unknown  terms  in  the  first  member  may 
be  united  into  one  and  the  coeflBcient  represented  by  a, 
and  all  the  known  terms  in  the  second  member  may  be 
represented  by  i.  If  a  is  negative,  the  equation  may  be 
divided  by  —  1.     Therefore, 

Prin.  83. — Bvery  pure  quadratic  equation  of  one  un- 
known quantity/  may  be  reduced  to  the  form  of  ax^'=^l,  in 
which  a  and  b  are  integral  and  a  positive. 

201.  Any  value  of  the  unknown  quantity  that  will 
satisfy  an  equation — that  is,  will  make  the  two  members 
equal — is  a  root  of  the  equation. 


PURE  QUADRATICS.  183 

202.  Take  the  equation  au?  =  1. 

Divide  by  a,  a;^  =  — . 

Take  the  square  root  of  both  members  [P.  69,  6], 


--/!■ 


Therefore, 


JPrin.  84:, — Every  pure  quadratic  equation  of  one  un- 
known quantity  has  two  roots,  numerically  equal,  hut  op- 
posed in  sign. 

2.   Solution  of  Pure  Quadratics. 


lUustratioiis.— 1.  Solve  V  +  5  =  2  a;^  _ 

-3. 

Solution :  Given           ?|^+    5  =  2a;»-3 

(A) 

Clearing  of  fractions,  3a;»  +  10  =  4a;«  —  6 
Transposing,                        —  a;»  —  —  16 
Dividing  by  —  1,                     a:^  =  16 
Taking  V,                               x-  ±4. 

(1) 
(3) 
(3) 

203.  Sometimes  equations  have  the  pure  quadratic  form, 
and  may  be  solved  as  such  when  they  are  not  really  such. 
Their  roots  may  not  be  numerically  equal. 

2.  Solve  (£±^  +  3a»  =  ii^'-2a».  (A) 

Suggestion. — 

Clear  of  fractions,  {x  +  a)«  +  12  a«  =  6  (a;  +  a)«  -  8  a«  (1) 

Transpose  terms,  —  5  (f  +  a)*  =  —  20  a«  (2) 

Divide  by  -  5,  (a;  +  a)«  =  4  a«  (3) 

Take  V,  x  +  a=±2a  (4) 

Transpose,  a;=  +  2o  —  a;or  —  2a  —  a      (5) 

Collect  terms,  x=z  a  or  —  3 a. 

EXERCISE    94. 

Solve  : 
1.  -^  +  -3-  =  lU  3.0;  +  -  =  - 

^--4      3-2a:2^5  3a:«-5_, 

^•""2        r~~-4      ^—^--^ 


184  ELEMENTARY  ALGEBRA. 

X     _x  —  2  X  a-\-x 


x-\-2        2x  '  a  —  x 


H) 


9 


6.  l^  +  ^l  =4ic2  10.  (Sx^-9y:=-a^ 

,   a      X  ,  '  ^+5      a;_3a:       17 

8.  --^ =  -  12.  [ax I  =-a^x^ 

X  c  \  x]       ^ 

13.  i(^  +  4)^  =  |(:.+  4)2-6 

14-  2-5  (5^  +  36)^  =  jg  (8  a^- 4)= 

15.  If  the  side  of  a  square  be  doubled,  its  area  will  be 
increased  75  square  rods.     What  is  the  side  of  the  square  ? 

16.  Four  times  one  number  equals  five  times  another,  and 
the  difference  of  their  squares  is  81.     Find  the  numbers. 

17.  A  man  bought  a  tract  of  land  for  $5000,  paying 
twice  as  many  dollars  per  acre  as  there  were  acres  in  the 
tract.     How  many  acres  were  there  ? 

18.  I  sold  a  horse  at  a  gain  of  $81,  and  thereby  gained 
as  many  per  cent  as  there  were  dollars  in  the  cost.  What 
was  the  cost  ? 

19.  Three  times  the  sum  of  two  numbers  equals  10  times 
the  smaller  number,  and  if  the  sum  be  multiplied  by  the 
greater,  the  product  will  be  630.     Required  the  numbers. 

20.  If  the  dimensions  of  a  certain  cube  be  quadrupled, 
the  entire  surface  will  be  increased  by  7290  square  inches  ; 
what  are  its  dimensions  ? 

21.  A  man  received  %  as  many  dollars  per  day  as  he 
worked  days ;  had  he  worked  only  ^5  as  many  days  and 
received  Yg  as  many  dollars  per  day,  he  would  have  re- 
ceived $26  less.     How  many  days  did  he  work  ? 


AFFECTED  QUADRATICS.  185 

Affected  Quadratics. 

I.   Definition  and    Principles. 

204.  A  quadratic  equation  which  contains  both  the  first 
and  second  powers  of  an  unknown  quantity  is  an  affected 
or  complete  quadratic  equation  ;  as, 

3ar  +  72:  =  15,  or— -5  =  — +6. 

206.  It  may  be  shown,  as  in  Art.  200,  that : 
Prin,  85. — Every  complete  quadratic  equation  of  07ie 
unknoiun  quantity  may  he  reduced  to  the  form  of  aoi?  -\- 
J)x  =  Cy  in  which  a,  h,  and  c  are  integral,  and  a  positive, 

206.  Take  the  equation  aa^ -\-l)x-=  c. 

Divide  by  a,  a^  H —  x=  -  , 

•^    '  ^  a         a 

h  c 

Put  p  for  —  and  a  for  — ,  x^  -\-px=z  n.     Therefore, 
^        a  ^         a  ^         ^ 

Prin,  86, — Every  complete  quadratic  equation  of  one 
unknown  quantity  may  he  reduced  to  the  form  of  x^  -\-px 
=  q,  in  which  p  and  q  may  he  integral  or  fractional, 
positive  or  negative. 


\  Since  p  and  q  may  be  either  positive  or  negatiye 
in  the  equation  x^  -^ p  x  =  q^  it  follows  that : 

Every  complete  quadratic  equation  may  he  reduced  to 
one  of  the  four  following  special  forms : 

1.  or -\-px= -\- q  3.  x^-\-px=— q 

2.  x^—px=-\-q  4:.  3r  —  px=  —  q 


2.    Solution  of  Numerical  Affected  Quadratics. 

208.  The  first  member  of  an  affected  quadratic  equation 
can  always  be  made  the  square  of  a  binomial  by  a  process 


186  ELEMENTARY  ALGEBRA. 

called  '^  completing  the  square,^''  then  the  square  root  of 
both  members  may  be  taken,  and  the  resulting  simple 
equation  solved. 

Illustration.— Solve  8  a;^  —  3  ic  =  26.  (A) 

Solntion  :  Multiply  by  2  to  make  the  first  term  a  perfect  square, 
16a:2-6a;  =  53  (1) 

Regard  \Qx^  —  Qx  as  the  first  two  terms  of  the  square  of  a  bino- 
mial, then  Vl6a;^  or  4ic,  is  the  first  term  of  the  binomial,  and  —Qx 
is  twice  the  product  of  the  two  terms;  therefore  {—Qx)-^{2  x  4a;), 

Q 

or  (—  6)  -4-  (3  X  4),  which  is  —-j-,  is  the  second  term  of  the  binomial, 

and  (  —  -J )  '  ^^  i7> '  ^^  ^^^  third  term  of  the  square  of  the  binomial. 
Add  this  to  both  members, 

16.^-6x-H^  =  53  +  ^  =  §  (2) 

Q  OQ 

Extract  V»     4  a;  -  -^  =  ±  ^  (3) 

13 
Transpose,  4  a;  =  8  or  — ^  (4) 

K 

Divide,  a;  =  3  or  —  1  -5- 

o 

209.  Hence  we  have  the  following  rule  : 

1.  Reduce  the  equation  to  one  of  the  typical  forms. 

2.  Multiply  or  divide  loth  memhers  of  the  equation  ly 
any  quantity  that  will  render  the  first  term  a  perfect  square. 

3.  Add  to  loth  members  the  square  of  the  quotient  ob- 
tained hy  dividing  the  coefficient  of  x  ly  twice  the  square 
root  of  the  coefficient  of  cc^,  to  complete  the  square. 

J/,.  Extract  the  square  root  of  both  members  and  solve 
the  resulting  simple  equation. 


Illustrations.— 1.  Solve  %a?  —  l%x  =  %. 

(A) 

Solution  :  Divide  by  3,              4  a;^  —  6  a:  =  4 

(1) 

-G-7f)'-l'     --e..|  =  44.- 

(3) 

Extract  the  V,                             2  a;  -  |-  =  ±  I 

(3) 

Transpose,                                            2  a;  =  4  or  —  1 

(4) 

Divide,                                                   a;  =  2  or  — -^ 

NUMERICAL  AFFECTED  QUADRATICS.         187 


2.  Solve  3  2;2  +  2a:  = 

:33. 

.»  +  |.= 

(A) 

Solntion  :  Divide  by  3, 

=  11 

(1) 

Add  (|  +  2vT)',  or  ( 

a;« 

4-1= 

=  "4 

_100 
9 

(3) 

Extract  V> 

1 

*+3  = 

-V" 

(3) 

Transpose, 

a;  = 

r3  or  — 

4' 

Therefore, 

Scholium,  1. —  When  the  coefficient  of  7?  is  1,  the  quan- 
tity to  be  added  to  both  members  to  complete  the  square  is 
the  square  of  half  the  coefficient  of  x. 


3.  Solve  3  3:^-5  a; 

= 

28. 

(A) 

Solution :  Multiply  by 

3, 

9a;»-15a:  =  84 

(1) 

-(;;.)'-(! 

)• 

9a;» 

25      ^,      25      361 
-15x  +  — =:84  +  -p  =  -p 
4                4        4 

(2) 

Extract  the  -y/. 

--|=±V' 

(3) 

Transpose, 

3a;  =  12  or  —7 

(4) 

Divide, 

a;  =  4  or  -2^. 
o 

Therefore, 

Scholiuni  2. —  When  the  equation  is  multiplied  through 
by  the  coefficient  of  a^,  the  quantity  to  be  added  to  both 
members  is  the  square  of  half  the  coefficient  of  x  in  the 
typical  equation.  This  method  is  generally  the  best,  as  it 
avoids  all  fractions  above  and  below  fourths. 


4.  Solve  2a:«  +  3a;  =  14.  (A) 

Solution  :  xMultiply  by  4  x  2,    16a;«  +  24  a;  =  112       •  (1) 

/   24   \« 
Add  (^-7--)  ,  or  3«,         lex*  +  24  a;  +  9  =  121  (2) 

Extract  the  V,  4a;  +  3  =  ±  11  (3) 

Transpose,  4  a;  =  8  or  —14       (4) 

Divide,  a;  =  2  or  —^2' 

Therefore, 


188  ELEMENTARY  ALGEBRA. 

Scholium  3, —  When  the  equation  is  multiplied  through 
by  four  times  the  coefficient  of  a:^,  the  quantity  to  he  added 
to  loth  members  is  the  square  of  the  coefficient  of  x  in  the 
typical  equation.  This  is  called  the  Hindoo  method  of 
completing  the  square.  It  avoids  all  fractions,  hut  often 
gives  rise  to  very  large  whole  numbers, 

EXERCISE    9S. 

Solve  : 
1.  a;2  -f-  2  a;  =  8  20.  16  rr^  —  16  o^  =  45 


2    .     r..,_  a  3  9 


2.  x^-'Zx  =  24.  2,7  10 

21.    X^  -\-—X=z 

Z.  x^-\-bx=-Q 

4:.  x^-dx=  —  20 

5.  ^2  _  3  ^  =  18 

6.  2;2  —  a:  =  20 

7.  :r2  -  11  a;  =  -  28 

8.  x^-^4:xz=60 

9.  x^-^x  =  66 

10.  x^  —  x  =  110 

11.  4:X--j-10x=-6 

12.  Sx^-'i!x  =  6 

13.  2a;2-7a;  =  30 

14.  6x^-\-dx  =  Sl 

15.  6x^-\-3dx=-28  29.  x^-2x=-~ 

16.  6x^-4.7x  =  (j3 

17.  6^^  +  190:= -15  30.  -^  +  ^^  =  2^ 

18.  2^:^-9^  =  35  ^  ^  ^ 

31. 


22. 

a;2  +  a;  =  8| 

23. 

^2-f  1      a;  +  2 

=4 

5       '       3 

24. 

2  2^2_3      ^_g 

7             14 

=4 

25. 

2:2-4a;=-l 

26. 

2;2-6a;  =  ll 

27. 

X         4 

28. 

ir+1          a; 

13 

u;          x-\-l 

42 

19.  9:^2  +  27 a:= -14  a^'+l      x-1      3^-1 

x^-{-x^l      a;^  -  a;  +  1  _  23 
^^'  ^       6  5  -  30 


LITERAL  AFFECTED  QUADBATICS,  189 

3.  Solution  of  Literal  Affected  Quadratics. 

ninstrations. — 1.  Solve  oc^ -\-ax  =  b. 
Solution :  C'omplete  the  square, 


Extract  the  V»        x-\-^  =  ±-^  V4&  +  a« 

Transpose,  a:  =  -|^  ±  ^V4&  +  a2= -^(a  T  V46  +  a2) 

2.  Solve  a7?-\-lx  =  c.  (A) 

Solution :  Multiply  by  a, 

a'a;' +  a&a:  =  ac  (1) 

Complete  the  square, 

6*               &«     4ac  +  6«  ^, 

a«a:«  +  a&a;  +  -j=ac  +  -j= ^ <^) 

Extract  the  V, 

aa;  +  |=±|V4ac  +  62  (3) 

Transpose,  a  a- =  —  -s-±:s-V4¥c  +  &«  ==—-^(&TV4ac  +  &2)       (4) 


3  -"  3 

1 


Divide  by  a,  a;  =  —  5— (J  T  V4  a  c  +  6*) 

EXERCISE    96. 

Solve  : 
1.  x^^'iax  —  3a^  4.  a^oi^  —  acx  =  'i(^ 


2.  a:2_3Ja:=:_2J2 

5.  x^-^(al)^y)x^ 

-al^ 

3.  a:^  +  m  a:  =  6  m^ 

6.  a:2_|_fl2_^^ 

7.2-^-^  =  3 
a        a; 

10.^  +  ^=^  +  ^ 

3,«  +  x           a      _^ 
a         a  —  X 

x         a  —  x 

x--b          h 
^'      a     "x-h 

12.  ^  +  l  =  a  +  i 

13.  (^  +  i)(x4-^)  =  — 

4. 

190  ELEMENTARY  ALGEBRA, 

(x -\- m)  (x  —  n)      imx  —  n^ 

14.  _  = _ 

15.  acx^—'bcx-\~adx--'bd=0 

16.  aJ)x^-{a^-¥)x-al  =  Q 


Equations  in  the  Quadratic  Form. 

Definitions. 

210.  When  an  equation  contains  two  and  only  two  ex- 
ponents of  the  unknown  terms,  and  one  of  them  is  twice 
the  other,  it  is  said  to  have  the  quadratic  form  ;  as, 

ic*+6a;2  =  16,  ax^^hx^^zc,  or 

{a-\-'b  xy  -[-p  {a-{-d  xf  =  c, 

211.  Any  equation  haying  the  quadratic  form,  whatever 
its  degree,  may  be  solved  by  any  of  the  methods  employed 
to  solve  an  affected  quadratic. 

lUustrations.— 1.  SoIyo  x^ -\- 6  a^  =  16,  (A) 

Solution :  Complete  the  square, 

^4  + 6^2 +  9 -25  (1) 

Extract  V,  a;^  +  3  =  ±  5  (2) 

Transpose,  x^  =  2  or  —  8  (3) 

Extract  ^/,  a:  =  ±  V  2,  or  ±  2  V^' 

2.  (x  +  4.Y-j-{x  +  4.Y  =  S^.  (A) 

Solution  :  Complete  the  square, 

{x  +  Af  +  i        )  +  -i-  =  4  (1) 

Extract  V,       (^^^  +  4)*  +  i  =  ±  2  (2) 

Transpose,  {x  +  4)^=1-^  or  —  2  -^  (3) 

4  . •      4  , 

Extract  ^,  a;  +  4  =  ±  4/1  J_  or  ±  4/  _  2^       (4) 

Transpose,  re  =  4  ±  y^  JL  or  4  ±  4/_2  — 


EQUATIONS  IN  THE  QUADRATIC  FORM.        191 

EXERCISE    97. 

Solve  : 

1.^  +  2x^  =  24  ,.81.+  !=^-^ 

2.  x«-9x'=-8  ^^ 

3.  a;»  +  a:*  =  6  8.  a:^  +  ^^^^  =  74  :c 

6.  ^-  +  31.^  =  32  10..^  +  ^  =  ^ 

11.  (a;  +  2)*  +  4(2:  +  2)2  =  21 

12.  (2a;  +  l)2  +  3(2a;+l)  =  70 

13.(.  +  iy-(.  +  l)  =  6 

14.  (x2^a;  +  2)2  +  a:2^^^2  =  6 

16.  {a?  -\-2xf  -2^(7^  -\-%x)  -\-VZO  =  0 

17.  (a:2_5^)2_8a;2  +  40a;  =  84 

15 


19.  2^|a:2_l\    _  11  (3^:2  _  2)  =  -10 

21.  a;'*  +  4j:''  =  12 

22.  ar^4-a:  +  -^-=3 

ar  '        'a:       4 


192  ELEMENTARY  ALGEBRA. 


Solution  of  Equations  by  Factoring. 

Illustrations. — 1.  Solve  4  a:^  =  1.  (A) 

Solution:  Transpose  1,  4a;2  — 1  =  0  (1) 

Factor,  {2x  +  \){2x-l)  =  0  [P.  39]  (3) 

Divide  by  (2a;  +  1),  2x  —  l  =  0 

Transpose  and  divide,  ^  =  o" 

Divide  (3)  by  (2x -  1),  2a;  +  1  =  0 

Transpose  and  divide,  x=:  —  — 

2.  Solve  a;2  +  5  a.-  +  6  =  0.  (A) 
Solution  :  Factor,  (a;  +  2)  (a;  +  3)  =  0  [P.  40]  (1) 
Divide  bya;  +  2,                           a;  +  3  =  0 

Transpose,  a;  =  3 

Divide  (1)  by  (a;  +  3),  a;  -  2  =  0 

Transpose,  a;  =  2 

3.  Solve  Qx^^llx-10  =  0,  (A) 
Solution  :  Factor,       (3  a;  -  2)  (2  a;  +  5)  =  0  [P.  41]  (1) 

(3) 

(3) 


Divide  by  (3  a;  — 2), 

2a;  4-5  =  0 

Transpose  and  divide. 

a;=- 

2- 

2 

Divide  (1)  by  (2  a; +  5), 

3a;-2  =  0 

Transpose  and  divide, 

2 
^=3 

4.   Solve  a?-l  =  (}. 

(A) 

Solution  :  Factor,    {x  -  1)  (a;^  +  a;  +  1)  =  0  [P.  44]  (1) 

Divide  by  (a;^  +  a;  +  1),  a;  —  1  =  0 

Transpose,  a;  =  1 

Divide  (1)  by  (a;  -  1),  a;^  +  a;  +  1  =  0  (2) 

Solve  (2),  a;  =  ^±iv^r3 

5.  Solve  re*  +  a.-^  +  1  =  0.  (A) 

Solution  :  Factor,  (a;^  +  a;  +  1)  (a;^  -  a;  +  1)  =  0  [page  84]      (1) 
Divide  by  (a;^  +  a;  +  1),  a;^  -  a;  +  1  =  0  (2) 

Solve  (2),  a;  =  4"  ±  T    ^ 


2-^2' 

Divide  (1)  by  {x^-x  +  1),       (a;^  +  a;  +  1)  =  0  (3) 

Solve  (3),  a;  =  -  i  ±  i  V^^ 


SOLUTION  OF  EQUATIONS  BY  FACTORING,    193 

6.  Solve  3^-a7r-a^x-\-a^  =  ^.  (A) 

Solution :  Factor  (A),   x^ {x- a)- a^{x- a)  =  0  (1) 

Factor  (1),  '  (x*  -  a«)  {x-a)  =  0  (2) 

Factor  (2X  {x  -  a){x  +  a)ix- a)=zO  (3) 

Divide  (3)  by  {x  -  a)  (.c  +  a),  x  —  a  =  0  (4) 

Transpose,  a:  =  o 

Divide  (3)  by  (a;  —  a)  (a;  —  a),  a:  +  a  =  0  (5) 

Transpose,  x=  —  a 

Divide  (3)  by  (a;  +  a)  (a;  -  a),  a:  —  a  =  0  (6) 

Transpose,  x  =  a 

7.  Solve  4a^  +  12a;3  +  29x2_|_30a;_j_2l.  (A) 

Solution :  By  extracting  the  square  root  of  the  first  member  we 
find  that  it  lacks  4  of  being  (2x^  +  Sx  +  5)«; 

.-.      (2a;^  +  3a;  +  5)«-4  =  0  (1) 

Factor,  (2^+3^+5"+  2) (2a;«+3a:  +  5- 2)  =  0  [P.  39]  (2) 


Collect  terras, 

(2a:*+3a:  +  ' 

7){2x*+Sx 

+  3)  = 

rO 

(3) 

Divide  (3)  by  2 

ia;«+3a;  +  7, 

2a;2+3a;  +  3  = 

=  0 

(4) 

Solve  (4), 

X  = 

=  - 

i(3±/^ 

-15) 

Divide  (3)  by  2 

la;»+3a:  +  3, 

2a;2+3a;  +  7  = 

=  0 

(5) 

Solve  (5X 

x  = 

--- 

^(3±V- 

-47) 

Kote.— This  equation  may  also  be  solved  by  adding  4  to  both 
members  and  extracting  V« 

EXERCISE    98. 

Solve  by  factoring : 

1.  ar~4  =  0  10.  9a;2  -  24a;  + 16  =  0 

2.  4a:2-9  =  0  11.  2x'-'7x-15  =  0 

3.  9a:2_4^2_o  12.  6a;2  _  13^  _|_  g  _.  q 

4.  2:2  _  1/^^  =  0  13.  S3^+Ux-15  =  0 
6.  a:2  4-7a;_|_io  =  0  14.  3x^ -{-Sx- 35  =  0 

6.  x^-6x-\-S  =  0  15.  ar^-8  =  0 

7.  ar  +  3a:-18  =  0  16.  2:^  +  1  =  0 

8.  42:2_2a;_i2  =  0  17.  a,'^  +  8  =  0 

9.  42r -I- 12  a; -1-9  =  0  18.  a:^  -  27  =  0 


194  ELEMENTARY  ALGEBRA. 

19.  a;^  +  27  =  0  22.  a;«  —  1  =  0 

20.  ic*  —  tt*  =  0  .  23.  x^  —  a^  =  0 

21.  x^  —  81  =  0  24.  82^  -  27 a^  =  0 

25.  :?:3  +  6  a;2  —  4a;  -  24  =  0 

26.  ic^  — a;2  — a;  +  l  =  0  28.  a;-^  +  3:z;2_^3a;  +  l  =  0 

27.  a;*  4-  a;2  +  1  =  0  29.  a;*  —  13  a;2  +  36  =  0 

30.  ic^-2a;3  +  3a;2-2a;  +  l  =  0 

31.  0:3  +  60:2  +  12 a;  +  8  =  0 

32.  0:*  — 8o:3  +  24o:2  — 32o:  +  7  =  0 

33.  {x-a){x-hy^x''-{a^h)x-\-ab  =  0 


Formation  of  Quadratic  Equations. 

I.   Principles. 

212.  If  we  solve  the  general  equation  x^-\-px=.q,  we 
will  find  the  roots  to  be  : 


W\/\ 


^P^  +  q  and 


The  sum  of  these  roots  is  —  ^  ; 

Their  product  is  —  -rp^  —  j  -jp^  +  2' )  =  —  S'* 

Therefore, 

rrin,  87^ — The  sum  of  the  two  roots  of  an  equation  of 
the  form  of  x^-\-px  =  q  equals  the  coefficient  of  a?,  with 
the  sign  changed. 

Frin,  88, — The  product  of  the  tivo  roots  of  an  equation 
of  the  form  of  x^-\-px  =  q  equals  the  absolute  term  with 
the  sign  changed. 


FORMATION  OF  QUADRATIC  EQUATIONS.      195 

2.   Examples. 

ninstrations. — 1.    Find   the  equation  whose  roots  are 
+  4  and  —  6. 

Solution  :  The  coefficient  of  a:  =  -  (4  —  6)  [P.  87]  =  +  2  ; 

The  absolute  term  =  -  (+  4  x  -  6)  [P.  88]  =  +  24 ; 

Therefore  the  equation  is  a;'*  +  2  a;  =  24. 

2.  Find  the  equation  whose  roots  are  2+V^  and  2— Vsl 
Solution  : 

The  coefficient  of  a;  =  -  {(2  +  V3)  +  (2  -  ^/3)}  [P.  87]  =  -  4; 
The  absolute  term    =  -  (2  +  V3 )  (2  -  VS")  [P.  88]  =  1 ; 
Therefore  the  equation  is  a;*  —  4  a;  =  1. 

EXERCISE    99. 

Form  the  equations  whose  roots  are  : 

1.  +  2  and  +  4  lO.  1  +  a/2  and  1  -  a/2 

2.  -  3  and  +  5  ii.  3  +  V2  and  3  -  a/2 

3.  _8and  +3  ^^   2a-dand2«  +  d 
4.-5  and  —  4 
6.  2«  and  a 

6.  3  jt?  and  -  2^  14.  -^  and  - 

7.  a  and  —  8  a  4  3 

.   T        1  -L  15.  TT  and  -r 

8.  a  +  c*  and  a  —  o  3  4 

9.  a^  +  ^  aiid  c?  —  W  16.  a  -f  2  m  and  a  —  2  m 


13.  a  4-  VJ  and  a—^/h 


Formation  of  Equations  by  Composition. 

Illustrations. — 

1.  Form  the  equation  whose  roots  are  +  2  and  —  2. 
Solution :  If  a;  =  +  2,    a:  -  2  =  0  (1) 

Ifa:=-2,    a:  +  2  =  0  (2) 

Multiplying  together  (1)  and  (2), 

(a;-2)(.c  +  2)  =  0  (3) 

Expanding,  x'  —  4  =  0 


196  ELEMENTARY  ALGEBRA, 

2.  Form  the  equation  whose  roots  are  —  4  and  -)-  7. 
Solution ;  If  a;  =  —  4,    a;  +  4  =  0  (1) 

Ifa;=  +  7,    iB-7  =  0  (2) 

(a:  +  4)(a:-7)  =  0  (3) 

Expanding,  a:^  —  3  a;  —  28  =  0 

3  2 

3.  Form  the  equation  whose  roots  are  —  -r  and  -. 

4  3 

Q 

Solution:         lix  =  —  -,    4a;=  —  3,  and  4ic  +  3  =  0        (1) 

2 

Ifa;  =  -g,        Zx  =  2,       and  3^-2  =  0        (2) 

(4a:  +  3)(3a;-2)  =  0  (3) 

or,  12a;2  +  a;— 6  =  0 

4.  Form  the  equation  whose  roots  are  -{-'^y  —2,  and  -1-3. 
Solution ;  It  x=  +  2,    {x  —  2)  =  0  (1) 

It  x  =  -2,    (a;  +  2)  =  0  (2) 

If  a;  =  +  3,    (a^  -  3)  =  0  (3) 

,-.      (a:-2)(a;  +  2)(a:-3)  =  0  (4) 

Expanding,  a:^  -  3  a;^  -  4  a;  +  12  =  0  (5) 

Observe  that  an  equation  always  has  as  many  roots  as  there  are 

units  in  its  degree. 

EXERCISE     lOO. 

Form  the  equations  whose  roots  are  : 

1.  -}-3  and  —  3 

2.  +  5  and  —  7 
3.-5  and  -\-  7 
4.-5  and  —  7 
5.  -|-5  and  +7 


6. 

2   ,3 
^and^   ' 

7. 

5  and  —  ^ 

8. 

3,2 
-^and-- 

9. 

+  1  and  -3 

0 

10. 

a  and  —  2  a 

11. 

2,  3,  and  —3 

12. 

3,  —  3,  and  4 

13. 

1,  2,  and  2 

14. 

3,  0,  and  1 

15. 

11    ^  1 

16. 

-,  -,  and  1 

17. 

0,  -,  and  -^ 

2     3         ,4 
18.  3,  4,  and  g 


MISCELLANEOUS  EXAMPLES,  197 

Miscellaneous  Examples. 

EXERCISE     lOl. 

Solve : 
1,  7^-a^  =  a^-  {x  -  af 

2ar       _       14  3  ,    6  3 


6. 


3a;+180      9a;-15  2a;  +  12^ic      2r2:  +  4 

10 10_  _  16 

12  +  3a;      3a;-12"~  9 


6  3:^4-10      2^:^  +  58_234 
2a;+14         2x-U   _        14 

8a 


Zx'-^ix      3  a^  -^Wx      3  a;2  _  219 

10.  ic(a:-l)  =  ^(a;2_|.22;) 

2  2a:-10  ,  2a:+6      ,  1 

11.  — * ' —  =  1  — 

3  8-a;    ^6-32:         3 

x-\-3  _3  —  2x      x  —  S 

^^'  x  +  'Z"  1-x  ^2-x 

X   ,  m      X  ,   n  o^o...! 

13.  —  +  —  =  -4--  16.  aar^  —  2a^x-]-a^=- 
m      X      n      X  a 

acT?      lex      ,  1  11.1 

14.  ax ^— =^ h     16. :r~. —  = T  +  - 

d  a  a  —  o-\-x      a      ox 

Find  the  approximate  values  of   x  in  the  following 
equations : 

l7.^-12a:  =  21  19.  1  -  ^  =  -  i 

2  X  of 

5  a;  +  l 


198  ELEMENTARY  ALGEBRA. 

^      39  a:^      '^Qx^      13  21  25       x 

■     28  42        14  10  -  2  a;  ~  14      14 

(5-^H^+5)_ll  5      3^+l_l 


Solve  : 

25.  ^^  +  A  =  «'  +  ^  26.  0:^  +  1  =  0 

27.  4i?;*  +  16.T3  +  16a;  +  4  =  57a;2 

28.  ^2_2:x;_2  +  --i^  =  0 

29.  Find  the  three  cube  roots  of  1. 
Suggestion. — Let  a;  =  V 1 ,  then  a;^  =  1,  or  a;*  —  1  =  0. 

30.  Find  the  three  cube  roots  of  —  1. 

31.  Find  the  four  fourth  roots  of  1. 

32.  Form  the  equation  whose  roots  are 

a,  —a,  h,  —  hy  c,  —  c. 


Examples  involving  Quadratic  Equations  of  One 
Unknown  Quantity. 

EXERCISE     102. 

1.  The  product  of  two  consecutive  numbers  is  156. 
"What  are  the  numbers  ? 

2.  The  sum  of  two  numbers  is  17,  and  their  product  is 
42.     Required  the  numbers. 

3.  The  difference  of  two  numbers  is  8,  and  their  prod- 
uct is  105.     What  are  the  numbers  ? 

4.  There  are  900  trees  in  an  orchard,  and  the  number 
in  one  row  exceeds  twice  the  number  of  rows  by  5.  How 
many  rows  are  there  ? 

5.  A  man  bought  some  cloth  for  $90 ;  had  he  bought 
15. yards  more  for  the  same  money,  he  would  have  paid  $1 
a  yard  less.     How  many  yards  did  he  buy  ? 


EQUATIONS  OF  ONE  UNKNOWN  QUANTITY.     I99 

6.  The  sum  of  two  numbers  is  30,  and  their  quotient  is 
the  less  number.     Required  the  numbers. 

7.  A  lot  that  is  2  rods  longer  than  wide  contains  48 
square  rods.     What  are  its  dimensions  ? 

8.  If  a  train  would  increase  its  speed  5  miles  an  hour, 
it  would  go  360  miles  one  hour  sooner.  What  is  the  rate 
of  the  train  ? 

9.  A  can  do  a  piece  of  work  in  2  days  less  than  B,  and 
they  together  can  do  it  in  2^5  days.  In  what  time  can 
each  alone  do  it  ? 

10.  One  pipe  can  fill  a  cistern  3  hours  sooner  than  an- 
other can  empty  it,  and  if  they  run  together  the  cistern 
will  be  filled  in  13  Ya  hours.  In  what  time  could  the  first 
fill  it  ? 

11.  A  certain  number  exceeds  its  square  root  by  30. 
Required  the  number. 

Suggestion. — Let  «'  equal  the  number. 

12.  If  the  circumference  of  a  wheel  were  increased  by  4 
feet,  the  wheel  would  make  110  revolutions  less  in  going  a 
mile.     What  is  the  circumference  of  the  wheel  ? 

13.  A  man  sold  a  horse  for  175,  and  thereby  gained  as 
many  per  cent  as  there  were  dollars  in  the  cost.  Required 
the  cost. 

14.  A  sold  his  farm  at  $48  an  acre,  and  thereby  lost 
one  half  as  many  per  cent  as  there  were  dollars  in  the  cost. 
Required  the  cost  per  acre. 

15.  A  man  increased  his  capital  stock  by  $500  without 
increasing  his  gain,  which  was  $500,  in  consequence  of 
which  his  rate  of  gain  was  lowered  5j^.  What  was  his 
original  stock  ? 

16.  How  much  must  be  added  to  both  the  length  and 
the  width  of  a  rectangle,  18  by  20  inches,  to  make  it  con- 
tain 483  square  inches  ? 


200  ELEMENTARY  ALGEBRA. 

17.  Eggs  rose  5  cents  a  dozen,  in  consequence  of  which 
3  eggs  less  could  be  purchased  for  25  cents.  What  was 
the  price  per  dozen  before  the  rise  ? 

18.  A  and  B  together  earned  $432.  B  earned  $6  a 
month  more  than  A,  and  the  number  of  months  they 
worked  was  one  third  of  the  number  of  dollars  A  earned 
in  a  month.  How  much  did  each  earn  per  month,  and 
how  many  months  did  they  labor  ? 

19.  A  boy  rowed  3  miles  down  a  river  and  back  again 
in  1  Ya  hour.  The  rate  of  the  current  was  2  miles  an  hour. 
Determine  his  rate  of  rowing  in  still  water. 

20.  A  and  B  are  320  miles  apart.  If  A  travels  8  miles 
a  day  more  than  B,  they  will  meet  in  one  half  as  many 
days  as  B  travels  miles  per  day.  How  far  does  each  travel 
per  day  ? 

21.  Twice  the  length  of  a  rectangle  equals  three  times 
the  width,  and  if  2  feet  be  added  to  the  length  and  3  feet 
to  the  width,  the  area  will  be  56  square  feet.  What  are 
the  dimensions  ? 

22.  A  and  B  have  each  a  debt  of  1150  to  pay.  A  pays 
13  a  week  more  than  B,  and  pays  his  debt  8V3  weeks 
sooner.     How  much  does  A  pay  per  week  ? 

23.  A  and  B  undertook  to  earn  1640.  A  earned  $8  a 
week  more  than  B,  and  the  number  of  weeks  required  was 
one  fourth  of  the  number  of  dollars  that  B  earned  in  a 
week.     What  were  the  weekly  wages  of  each  ? 

24.  Around  a  flower-bed,  18  feet  by  12;  is  a  gravel-walk 
whose  area  equals  that  of  the  flower-bed.  What  is  the 
width  of  the  walk  ? 

25.  A  farmer  sold  7  pigs  and  12  lambs  for  $50,  and 
found  that  he  had  sold  3  more  pigs  for  $10  than  he  sold 
lambs  for  $6.     Required  the  price  of  each. 

26.  One  person  husked  48  shocks  of  corn  in  a  day ; 
another  husked  the  same  number  two  hours  sooner,  and 


EQUATIONS  OF  TWO   UNKNOWN  QUANTITIES,    201 

husked  2  shocks  per  hour  more  than  the  first.     How  many 
shocks  per  hour  does  each  husk  ? 

27.  A  and  B  were  engaged  at  different  rates  of  wages. 
A  worked  a  certain  number  of  days  and  received  $24,  and 
B,  who  worked  6  days  fewer,  received  $13  Vg.  If  A  had 
worked  6  days  fewer  and  B  6  days  more,  they  would  have 
received  the  same  sum.     How  many  days  did  each  work  ? 


Quadratic  Equations  of  Two  Unknown  Quantities. 

Definitions. 

213.  A  quadratic  equation  of  two  unknown  quantities 
is  complete  when  it  contains  all  the  second  degree  and  all 
the  first  degree  terms  possible  ;  as, 

ax^-{-l)xy-\-cy^-{-dx-\-ey  +/  =  0. 

214.  A  quadratic  equation  of  two  unknown  quantities 
is  pure,  or  homogeneous,  if  all  the  terms  containing  un- 
known quantities  are  of  the  second  degree  ;  as, 

aa? -{-hxy -\-cy'^  =^  d. 

215.  Any  equation  containing  two  unknown  quantities 
is  symmetrical  if  the  unknown  quantities  may  change 
places  without  destroying  the  equation  ;  as, 

2a;«  +  3a:y  +  2^2^12  and  2y^ -{-Zyx-^23?  =  \%. 

216.  The  solution  of  two  simultaneous  quadratic  equa- 
tions of  two  unknown  quantities  often  involves  the  solu- 
tion of  a  bi-quadratic  equation. 

Ulnstration. — 

Given  I  ^'  ~  y^ ""  I        (,t!  \  to  find  x  and  y. 
U  -  /  =  3        (B)  j 

Solution  :  Transpose  (A),  y  =  re'  -  2  (1) 

Square  (1),  y»  =  a:*  -  4  j:«  +  4         (2) 

Substitute  (2)  in  (B),     x-*  —  4a;»  —  a;  =  —  7,  a  bi-quadratic. 


202  ELEMENTARY  ALGEBRA. 

Solvable  Classes. 

I.  When  one  equation  is  of  the  first  degree  and  the 
other  of  the  second  degree,  they  are  solvable  as  quadratics. 


niustrations.— 1.  Solve  P  ^  ^  ^  ~  f 
Solution:  Transpose  (A),                2x  =  S  —  y 

(A)) 
(B)f 
(1) 

Divide  (1)  by  2,                                 ^  =  ^i^ 
Substitute  (2)  in  (B),  ^^~^'-2y^  =  2 

(3) 
(3) 

Clear  of  fractions,         Sy  —  2/^  —  22/^^  =  4 
Rearrange  terras,                  32/2_8y  =  —  4 
Complete  the  square,  92/^  —  24?/  +  16  =  —  12  +  16 
Extract  the  V,                         3  2/  —  4  =  ±  2 

=  4 

(4) 
(5) 
(6) 
(7) 

o 

Transpose  and  divide,                       y  =  2ov-^ 

(8) 

Substitute  (8)  in  (2),                        x  =  i(8-2)  or 

K- 

-1)  <«) 

Reduce  (9),                                        a;  =  3or3|- 

o 

217.  Many  equations  of  this  class  may  be  solved  by 
more  elegant  methods. 

^-  ^°'^«   1      xy  =  n  (B)f 

Square  (A),  x"^  +  2xy  +  y^  =  A9  (1) 

Multiply  (B)  by  4,  4xy  =  48  (2) 

Subtract  (2)  from  (1),  x^  —  2xy  +  y^=l  (3) 

Extract  V>  x  —  y=±l  (4) 

Add  (4)  to  (A),  2a;  =  8or6  (5) 

Divide  by  2,  a;  =  4  or  3 

Subtract  (4)  from  (A),  22/  =  6or8  (6) 

Divide  by  2,  ?/  =  3  or  4 

Therefore  y  =  S  when  a;  =  4,  and  y  =  4  when  x  =  d. 

In  a  similar  manner  may  be  solved  equations  of  the 
form  of : 

ix~y  =  a)  {x^  +  f  =  a)  ix^-{-i/  =  a) 

\      xy=h)'       \         xy=b)'       \    x  ±  y  =  h  ) 


EQUATIONS  OF  TWO   UNKNOWN  QUANTITIES.    203 


EXERCISE     103. 


1.  \x-^y=    9) 
\      xy  =  %0) 

2.  \x-y=    6) 

4.    (a:«  +  /  =  73) 
\x^y  =^1%\ 


\x^^4.f 


25 
6 


a:2  +  r  =  52 
cc?/  =  —  24 


1 

11.  (  a;2-9«/2  =  16  \ 

\x  -Zy  =    2) 

12.  (  2  a;  +  3  3/  =  40  ) 
(  a;?/ =  50! 

13.  {4:a^-9y 


{3^-\-y^=50) 
\x  —y  =  10  ) 


X  -\-y  =7       f 
^  +y  =a  -\-l 


14. 


15. 


-  9  /  =  108  ) 
-Sy  =-6f 

(a:  +  2y  =  17) 
|a;«  +  /  =  61  j 


(30:2  =  4/ +  48 


I  a:^  -  /  =  «2  _  j2  I 

(^  +?/  =«  +5   j 

(  a;  —  ?/  =    4  i 


-I 


17. 


a;  +  3«/=    2  ) 
x'-{-xy  =  ^8  \ 

i  3x-7y=-5 
\  xy  —  y^  =  25 

7 


-  +  ^ 

1       1  __  25 


9.    r_l  _  j^_  5  ]  18. 

1  _!__  J^ 
a;       y  ""  6 

218.  Some  equations  of  a  higher  degree  may  be  reduced 
to  this  class  by  division. 

niustratioiL-Solve  K  "  2/'  =  56  (A)  ) 

\x  -y  =    2  (B)  j 

Solution :  Divide  (A)  by  (B),   x^+xy  +  y^  =  28  (1) 

Square  (B),                           x^  —  2xy  +  y^  =  4:  (2) 

Subtract  (2)  from  (1),                          3xy  =  24:  (3) 

Divide,                                                    xy  =  S  (4) 

Add  (4)  to  (1),                     x^  +  2xy  +  y'  =  3Q  (5) 

Extract  the  V»                                     x  +  y=±G  (6) 

Add  (B)  and  (6),  and  divide,                   a;  =  4  or  -  2  (7) 

Substitute  (7)  in  (4),  and  divide,             y  =  2  or  —  4  (8) 
Therefore,  when  x  =  4,  y  =  2,  and  when  x  =  —  2,  y=  —4, 


204  ELE3IENTARY  ALGEBRA. 

In  a  similar  manner  may  be  solved  equations  of  the 
form  of  : 

io^-{-if=:a)        {a^-f  =  a\        {  x^ -{- a:^  y"^ -{- f  =  a  ) 
\x-^y=b\'    \x'±y^  =  b\'    \x'±x  y  -{-y'=b\ 


EXERCISE     104. 


Solve  : 

{x'-^y^  =  35)  7.    (  c^-^  +  27y3  =  243  ) 

\x  -\-y  =    6\  \x  +    Sy  =      9\ 

ix^-y^  =  61)  e.    iSa^-y^  =  9S) 

\x  -y  =    l\  \2x  -y  =    2\ 

{a^-f  =  80)  9.    (  8a;3  +  27^3  =  35) 

\a^-y^=    8f  |2^  +    3^  =    5S 

j  ic*  -  /  =  -  65  )  10.    (  :c*  -  16/  =  80  ) 

\x^  +  y^=      13  \  \x^-    4.y^=    s] 

{x'-\-a^y^-\-f=:4:Sl)  ii.    {Sla^-16f=176) 

\o^-x  y  -^y^=    13)  (    9a;3+    4/=    25  f 

''^  +  ^^/  +  «/*  =  21  I  12.   j  16ic4+4a.V+/=91  ) 

^  +  ^  y  +/=    3  )  (    ^.x'-^xy  -]-y^=   7  j 


219.  Sometimes  there  is  a 

common  factor  in 

the  first 

smbers  of  two  equations  that 

may  be  removed  by 

'  division. 

lUustration.— Solve  \  ^^  ~ 

\xy- 

■y^  =  2 

(A)) 
(B)i 

Solution  :  Factor  (A)  and  (B),    {x -^  y){x  — y)  =  ^ 

(1) 

y{x-y)  =  2 

(3) 

Divide  (1)  by  (2), 

x  +  y      5 
y     ~  2 

(3) 

Clear  of  fractions, 

2x  +  2y  =  5y 

(4) 

Transpose, 

,2x  =  3y 

(5) 

Divide, 

3 

(6) 

Substitute  (6)  in  (2), 

jy'  =  ^ 

(7) 

Reduce, 

y=±2 

(8) 

Substitute  (8)  in  (6), 

x=±3 

(9) 

EQUATIONS  OF  TWO  UNKNOWN  QUANTITIES.     205 

EXERCISE     lOS. 

Solve  : 

1.  {x^   -y^=12)  4.    {4:0^  -9/= -108) 
\xi/-\-7/  =  12^  \2xi/-Sf=-   24S 

2.  {x^-\-2xy  =27)         5.    {  4ta^-{-8xy-\-3y^=96  [ 
\x'-{-3xj/-\-2f=D4:\  \  2a^-{-xy=^S') 

3.  {x^-\-3xy-{-2f=12)        6.    (16a;2-9/   =319) 
\3^-^4.xy-}-dy'-=16]  \    Sa^ -^6xy  =  2do\ 

7.  i  4ar  +  4a;y  +  «/2  =  169  ) 
t  2a:y  +  2/'=    39  f 

8.  {a^-3xy-4y^   =-150) 
(  2a^-Sxy=-160) 

1  9a:y-6r  =  30) 

10.    (  Ga.'2  +  19a;y  +  15/  =  40) 
\6x^-xy-15y^=-lo\ 

220.  Sometimes  one  or  both  equations  have  the  quad- 
ratic form,  or  may  be  reduced  to  the  quadratic  form, 
ninstrations. — 

•  ^    ^  I  x-y  =   i  (B)f 

Solution  :  Complete  the  square  in  (A), 

1  1       81 

(X  +  yy  +  (X  +  y)  +  j  =  20  +  j  =  ^  (1) 


Extract  the  7, 

x  + 

2^+2  =  ^2 

(2) 

Transpose, 
Add  (B)  to  (3), 

a;  +  y  =  4or  — 5 
2a;  =  8  or  —1 

(3) 
(4) 

Divide, 

a;  =  4or  -^ 

(5) 

Subtract  (B)  from  (3), 

2y  =  0  or  —  9 

(6) 

Divide, 

y  =  0  or  -4-^ 

Therefore,  when  a;  =  4,  y  =  0,  and  when  x=  —  ^,  y=  —  4 
10 


206  ELEMENTARY  ALGEBRA. 

2.  Solve  1^^  +  ^^  +  '  +  ^  =  ''  (^H 

Solution:  Add  (B)  to  (A), 

x^  +  2xy  +  y^  +  2x  +  2y  =  m  (1) 

Factor,                 {x  +  yf  -\- 2{x  +  y)  =  63  (2) 
Complete  the  square, 

{X  ■\-  yf  +  2{x  +  y)  +  \  =  U  (3) 

Extract  the  V»                   a;  +  y  +  1  =  ±  8  (4) 

Transpose,                                 a;  +  y  =  7  or  —  9  (5) 
Substitute  (5)  in  (A)  and  (B),  and  transpose, 

a;2  +  2/'  =  25  or  41       -  (6) 

2xy  =  M  or  40  (7) 

Subtract  (7)  from  (6),  x^—2xy  +  y^=\ovl  (8) 

Extract  the  V»                          x  —  y=±\,oY±\  (9) 

Add  (9)  and  (5),                           2  a;  =  8  or  6,  —  8  or  —  10  (10) 

Divide,                                              a;  =  4,  3,  -4,  or  -5  (11) 

Subtract  (9)  from  (5),                  2  3/ =  6  or  8,  —  10  or  —  8  (12) 
I>ivide,                                              y  =  3,  4,  -  5,  or  -  4 


EXERCISE     106. 

Solve  : 

1.  i  a;2/  +  a:i/  =  42)  8.    (  a;2  +  /=100| 
I          x-^y=    5)                      \2xy-{-x^y=llo\ 

2.  i    X^     ,     2X  .    4   )  9.       [    X^  7/2 


^^r^^_^y^27 


"^  y       ^9^  \y''^x^~^l}  +  '^-^ 


[      x-y  =  l     )  (  ^.^^2^20 

3-     (  (^  +  2/)'  +  ^  +  «/  =  56|        10.    j  a^J^y^  =  5S 

\  xy  =  10\  \xy-x  +  y  =  25 

4.    Ua^-^y^y^a^-\.y2  =  so^ 
\  a;2  ~  /  =    3  ! 

6-    {{x-{-yy-\-2x  +  2y  =  80) 
I     (x-yy-{x-y)=    6f 

6-    ((^  +  #-3(^  +  ^)  =  575) 
1  icy  =  150t" 


EQUATIONS  OF  TWO  UNKNOWN  QUANTITIES.    207 


II.  Two  homogeneous  equations  of  the  second  degree 

may  be  solved  by  putting  y  =zvx  when  they  can  not  be 

more  easily  solved  otherwise. 

^+    xyJty'=      28        (A)   ) 
/=-28         (B)   ] 


UluBtration.— Solve  |  !I  "^  «  ^  ^  "^ 

Solution:  Substitute  vx  tor  y  in  (A)  and  (B), 
a;s  +  va:«  +  t;«a;«  =  28 

Factor  (1)  and  (2)  and  divide, 

x^  (1  +  f  +  v^)  =    28 
a;8(l_2f-v»)  =  -28 
or,  1  +  V  +  v^  _ 


l-2v 


Clear  of  fractions, 

Transpose, 

Substitute  in  (3), 

Divide, 

Extract  the  >/, 

Substitute  (10)  in  y  =  vx, 


l  +  v  +  v^  =  —  l-\-2v  +  v^ 
v=   2 
7a;«  =  28 
x^=   4 
x=  ±2 
y  =  2x(±2)  =  ±4. 


(1) 
(2) 

(3) 
(4) 

(5) 

(6) 
(7) 
(8) 
(9) 
(10) 


Solve 


EXERCISE     107. 


3. 


1= 


a^  -\-y^  =  6 
2a^-\-xy-\-y^  =  S 


2.    (       a^-xy-\-y'^=12) 


a^  —  xy  =  —  —z 
^  16 


ia?JrxyJ^y^  =  2S\ 
\       2a:2_|_3^2^44f 


j 


5a:2-3y2=_63 
a*  +  a;y  =  27 


x^-\-2xy-\-%y^  =  b 
2a^  +  5/  =  7 

3/^ 


]  \      f-x^=    8f 


M 


2x2-a;y-y2=  _40] 
a^  +  /  =  40       f 

7?  —  xy  —  y^  =  —  125  ) 
a^-\-2xy=zl25       f 


).    J2:«  +  3a;y  +  2y2  =  40  ) 


ELEMENTARY  ALGEBRA, 


Solve  : 


Miscellaneous  Examples. 

EXERCISE     108. 


6. 


8. 


10. 


xy-y' 

=  11) 


(  x-^y=    2) 


x  —  ^yz=z 

x-\-y  =  a  —  h 
xy—  —  ah 

x^  +  y^  =  a^  i 
xy  =  l   f 

I  a^  +  2/=^  =  126  ) 
(    ^+y=      6f    • 

(    a?-xy-\-y^=   21 


u-^+^z+^z' 


133 
19 


a;3  +  1/3  =  7  (a;  +  ?/) 
x-y=l 

x^-{-xy  =  A:% 
y^-\-xy  =  lQ 


11. 


12. 


13. 


14. 


15. 


16. 


17. 


18. 


20. 


I 

S^(^  +  «/)  +  2/(^  +  «/) 

(  xy{x-\-y 


=4 

=  80 


a; 


y 


x-y      x-^y 
2-i-Sxy 

x^-\-y^-l  = 
xy(xy-^l)  = 

x^y^x-^yY  = 
a^-\-y^  = 

1 


=  dx) 

2xy) 

42      f 

2916  ) 
2xy^ 

1   ] 


19.    Ja;2  +  /  +  2^  +  2^  =  50) 

xy-i-x-{-y  =  23^ 


81 
180 


21.    ^  :z;4-?/4-2;=    6 

-l  xy  -\-xz~\-yz=  11 
(  a;  +  2;  =  ?/^ 


EQUATIONS  OF  TWO  UNKNOWN  QUANTITIES.    209 

Examples  involving  Quadratic   Equations  of  Two 
Unknown  Quantities. 

EXERCISE     109. 

1.  The  sum  of  two  numbers  is  13,  and  their  product  is 
40.     Find  the  numbers. 

Suggestion. — Let  x  equal  one  number  and  y  the  other. 

2.  Tiie  difference  of  the  squares  of  two  numbers  is  40, 
and  their  sum  is  10.     Find  the  numbers. 

3.  A  man  has  two  fields  in  the  form  of  squares,  contain- 
ing 16,400  square  rods,  and  the  one  is  20  rods  longer  than 
the  other.     Required  the  length  of  each. 

4.  The  difference  between  two  cubical  blocks  of  marble 
is  152  cubic  feet,  and  the  difference  of  their  lengths  is  2 
feet.     Required  the  length  of  each. 

5.  A  has  a  rectangular  field  containing  240  perches,  and 
a  square  field  containing  676  perches.  If  the  side  of  the 
square  field  is  equal  to  the  diagonal  of  the  rectangular  one, 
what  are  the  dimensions  of  each  ? 

6.  A  man  bought  a  number  of  horses  for  $3600 ;  had 
he  bought  five  more  at  15  apiece  less,  they  would  have  cost 
him  $225  more.    How  many  did  he  buy,  and  at  what  price  ? 

7.  A  and  B  each  worked  as  many  days  as  he  received 
dollars  a  day,  and  together  received  $89.  Had  A  worked 
as  many  days  as  B,  and  B  as  many  as  A,  they  would  have 
received  only  $80.  How  long  did  each  labor,  and  what 
did  he  receive  per  day  ? 

8.  The  product  of  two  numbers  exceeds  the  square 
root  of  the  product  by  30,  and  the  quotient  exceeds  the 
square  root  of  the  quotient  by  74.    What  are  the  numbers  ? 

9.  There  is  a  number  consisting  of  two  digits  ;  the  sum 
of  the  squares  of  the  digits  exceeds  their  product  by  21, 
and  if  9  be  added  to  the  number  the  digits  will  change 
places.     Required  the  number. 


210  ELEMENTARY  ALGEBRA. 

10.  A  man  and  boy  worked  at  one  time  as  many  weeks 
as  the  man  earned  dollars  a  week,  and  received  $700 ;  at 
another  time  as  many  weeks  as  the  boy  earned  dollars  per 
week,  and  received  1525.  How  much  did  each  earn  per 
week  ? 

11.  A  man  has  a  rectangular  lot  containing  1  acre ;  if 
it  were  4  rods  longer  and  4  rods  narrower,  it  would  con- 
tain only  %  of  an  acre.  What  are  the  dimensions  of  his 
lot? 

12.  The  fore- wheel  of  a  carriage  makes  88  revolutions 
more  in  going  a  mile  than  the  hind-wheel ;  but  if  the  cir- 
cumference of  the  fore-wheel  be  diminished  1  foot,  it  will 
make  146%  revolutions  more  than  the  hind- wheel.  What 
is  the  circumference  of  each  wheel  ? 

13.  A  and  B  each  invested  $182  in  wheat,  A  receiving 
10  bushels  more  than  B.  Had  A  paid  5  cents  a  bushel 
more  for  his,  and  B  5  cents  a  bushel  less  for  his,  they 
would  have  received  the  same  amount.  At  what  prices 
did  they  buy  ? 

14.  A  man  sculled  24  miles  down  a  river  and  back  again. 
He  found  that  it  took  him  8  hours  longer  to  return  than 
to  go,  and  that  his  rate  down  was  3  times  his  rate  return- 
ing. What  was  his  rate  of  sculling  in  still  water,  and 
what  was  the  rate  of  the  current  ? 

15.  The  area  of  a  rectangle  is  4  feet  less  than  the  area 
of  a  square  of  equal  perimeter,  and  the  length  is  Vg  of  the 
breadth.     Required  the  side  of  the  square. 

16.  The  greater  of  two  numbers  divided  by  their  sum, 
added  to  the  smaller  divided  by  their  difference,  gives  3Y7, 
and  the  difference  of  their  cubes  is  4625.  Required  the 
numbers. 

17.  The  difference  between  the  hypotenuse  and  the  base 
of  a  right  triangle  is  6,  and  the  difference  between  the 
hypotenuse  and  the  perpendicular  is  3.  What  is  the 
length  of  each  side  ? 


NEGATIVE  RESULTS,  211 

18.  A  man  invested  equal  sums  of  money  in  6^  and  7^ 
stocks,  paying  $10  a  share  more  for  the  latter  and  receiving 
10  shares  less.  The  income  of  the  latter  was  $20  more 
than  of  the  former.  What  was  the  sum  invested,  and  the 
price  of  tlie  shares  ? 

19.  A  drover  sold  10  horses  and  7  cows  for  $800.  He 
sold  5  cows  more  for  $160  than  he  did  horses  for  $198. 
At  what  price  did  he  sell  each  ? 

20.  A  and  B  start  together  on  a  journey  of  36  miles. 
A  travels  one  mile  per  hour  faster  than  B,  and  arrives  3 
hours  before  him.     Find  the  rate  of  each. 

21.  Two  partners  gained  $140  by  trade.  A's  money 
was  in  trade  3  months,  and  his  gain  was  $60  less  than  his 
stock  ;  B's  money  was  $50  more  than  A's,  and  was  in  trade 
6  months.     What  was  A's  stock  and  what  was  B's  gain  ? 


Negative  Results. 

221.  Some  questions,  evidently  intended  to  be  taken  in 
an  arithmetical  sense,  give  rise  when  solved  by  algebra  to 
negative  results.     How  shall  these  results  be  interpreted  ? 

1.  A  negative  result  may  arise  from  an  erroneous  state- 
ment of  a  condition. 

niustration. — A  certain  number  increased  by  5  equals 
%  of  the  number,  diminished  by  3.    Required  the  number. 

Solution :  Let  x  =  the  number ; 

Q 

then    x-\-^=.-rX  —  ^i 
4 

or,  4a:  +  20  =  3a:  — 12; 

whence,  a;  =  —  32. 
Tn  an  algebraic  sense  this  result  may  be  verified,  but  in  an  arith- 
metical sense  it  is  meaningless.  The  condition  of  the  question  as 
stated  is  erroneous.  If  it  be  modified  to  read,  "A  certain  number 
diminished  by  5  equals  2/4  of  the  number,  iticreased  by  3,"  the  result 
will  be  32,  which  is  consistent  and  intelligible. 


212  ELEMENTARY  ALGEBRA. 

2.  A  negative  result  may  arise  from  an  erroneous  state- 
ment of  a  question, 

niustration. — A  man  is  40  years  old,  and  his  brother  is 
25.     When  will  he  be  twice  as  old  as  his  brother  ? 

Solution  :  Let  x  =  the  number  of  years  hence ; 

then  40  +  a:  =  2 (25  +  a;)  =  50  +  22; ; 
whence,      a:  =  —  10. 

In  an  arithmetical  sense  this  result  is  not  intelligible.  If  the 
question  be  asked,  "  When  was  he  twice  as  old  as  his  brother  ? "  the 
result  will  be  10  years,  which  will  satisfy  the  question. 

S.  A  negative  result  may  arise  from  an  erroneous  sup- 
position made  in  the  solutio7i  of  a  question. 

lUustration. — A  man  sold  his  horse  for  $80 ;   had  he 
sold  him  for  $40  more,  he  would  have  gained  20^.     lie- 
quired  his  gain  or  loss. 
Solution  :  Let  x  =  his  gain ; 

then  80  —  a:  =  the  cost, 

and        -^  (80  —  x)  =  the  gain  by  second  condition ; 
whence  -^  (80  —  a:)  =  the  selling-price  by  second  condition. 

|-(80-a;)  =  $120, 

and  x=  —  $20. 

In  an  algebraic  sense,  gaining  —  $20  is  equivalent  to  losing  $20. 
Had  X  been  assumed  equal  to  the  loss,  the  result  would  have  been  $20. 

J/..  Negative  results  in  examples  involving  quadratics 
are  generally  the  numerical  equivalents  of  positive  results 
in  analogous  examples. 

Illustrations. — 1.  A  man  has  a  square  board,  such  that 
the  number  of  inches  in  length  added  to  the  number  of 
square  inches  in  the  area  equals  12.     Kequired  the  length. 
Solution :  Let  x  =  the  length ; 

then        x^  =  the  area, 
and  a:^  +  a;  =  12 ; 
whence     x  =  S  or  —4. 
x  =  S  satisfies  the  question  as  stated,    a;  =  —  4  indicates  that  the 
number  of  inches  should  be  arithmetically  subtracted  from  (algebraic- 
ally added  to)  the  area. 


NEGATIVE  RESULTS.  213 

2.  If  to  280  more  than  the  square  of  my  age  you  add 
34  times  my  age,  the  result  will  be  zero.    Required  my  age. 

Solution :  Let  a;  =  my  age ; 

then  x^  +  280  +  34a:  =  0; 
or,  a;*  +  34^= -280; 

whence  a;  =  —  14  or  —  20. 

No  value  of  x  will  arithmetically  satisfy  the  question  as  stated. 
But,  since  x^  is  positive  and  34  a;  is  negative  for  these  values  of  a:, 
the  analogous  question,  "  If  from  280  more  than  the  square  of  my  age 
you  subtract  34  times  my  age,  the  result  will  be  zero;  required  my 
age,"  will  be  satisfied  by  a:  =  14  or  20. 

EXERCISE     no. 

Solve  and  interpret  the  results  of  the  following  examples : 

1.  What  number  increased  by  7  equals  5  ? 

2.  12  diminished  by  what  number  equals  20  ? 

3.  A  man  is  40  years  old  and  his  son  is  20.  In  how 
many  years  will  the  father  be  272  times  as  old  as  the  son  ? 

4.  A  line  40  feet  long  was  cut  into  two  parts,  sucli  that 
one  part  increased  by  20  feet  equaled  the  other  part  dimin- 
ished by  30  feet.     Required  the  length  of  each  piece. 

5.  The  numerator  of  a  fraction  is  2  greater  than  the 
denominator,  and  if  9  be  added  to  both  terms  the  result 
will  equal  2.     What  is  the  fraction  ? 

6.  Two  thirds  of  A's  age  increased  by  12  years  equals 
^5  of  his  age.     What  is  his  age  ? 

7.  The  square  of  a  number  diminished  by  the  number  is 
12.     Required  the  number. 

8.  The  product  of  the  sum  and  difference  of  two  num- 
bers is  17,  and  one  of  the  numbers  is  9.  What  is  the  other 
number  ? 

9.  A  garden,  40  yards  long  and  30  yards  wide,  has  a 
gravel-walk  along  its  perimeter  that  occupies  "/is  of  the 
garden.     Required  its  width.        Ans.,  2V2  yd.  or  32 y^  yd. 

Interpret  the  meaning  of  32  V2  yards. 


CHAPTER  VI. 

EXPOJ^EJ^TS,  RADICALS,  AJfD 
IJfEQUALITIES. 


Fractional  ^and  Negative  Exponents. 

Principles  and  Applications. 

222.  We  learned  [P.  75]  that  dividing  the  exponent  of 
a  factor  by  the  index  of  a  root  extracts  the  root  of  the 
factor.  If  this  principle  be  accepted  as  general  in  its  char- 
acter, and  applied  when  the  exponent  of  the  factor  is  not 
divisible  by  the  index  of  the  root,  the  result  will  be  a 
quantity  with  a  fractional  exponent.     Thus, 

V«^  =  a^,  read  a,  exponent  two  thirds. 

223.  A  fractional  exponent,  from  the  nature  of  its 
origin,  denotes  a  root  of  a  power,  the  numerator  being 
the  exponent  of  the  power,  and  the  denominator  the  index 
of  the  root. 

224.  Since 


«»  z=  Vfl^"*  [223]  =  Va  X  aX  a  X to  m  factors  = 

Vax  Vax  VaX...,  to  m  factors  [P.  76]  =  (V^)",  it 
follows  that  a  fractional  exponent  may  also  be  regarded  as 
denoting  a  power  of  a  root,  the  numerator  still  being  the 
exponent  of  the  power,  and  the  denominator  the  index  of 
the  root. 


FRACTIONAL  AND  NEGATIVE  EXPONENTS,    215 

SIGHT      EXERCISE. 

Name  the  value  of  : 

1.  8^  3.  (a^)^  5.  (-ar^)^  7.  32^ 

2.  (-8)^  4.  10^  6.  27^^  8.  {b')^ 

'■  ig    "■  (i)'    '■•  (I)' 

Name  the  equivalents  of  the  following  quantities  : 


12.  Va 

15.    V^ 

18.  X^ 

21.  (Va)3 

13.  Va 

16.  (Vay 

19.  X^ 

22.  (\/'a)5 

14.  7^ 

17.  4* 

20.  3« 

23.  (Viy 

225.  Since  a^^  =  'V^  =  i^  Va«  =  V?=  «^,  it  follows 
that  a^  and  aA  are  equivalent.     Therefore, 

^Hw.  89, — Multiplying  or  dividing  both  terms  of  a 
fractional  exponent  by  the  same  quantity  does  not  change 
its  value. 

226.  Since  a^  =  a-^  [P.  891,  and  a^  =  a^  [P.  89], 
aixai  =  a^Xa^  =  (Vaf  X  (V^Y  =  ('V^)"  =  a+^  = 
a^+^,  it  follows  that, 

Prin.  90* — The  exponent  of  a  factor  in  the  product 
equals  the  sum  of  the  exponents  of  the  same  factor  in  the 
multiplicand  and  multiplier,  when  the  exponents  are  posi- 
tive fractions. 

227.  Since  a?  X  a^  =  a^  [P.  90],  it  follows  that  a^  -r-  a* 
=  at=:fljH— J.     Therefore, 

Prin.  91, — The  exponent  of  a  factor  in  the  quotient 
equals  the  exponent  of  the  same  factor  in  the  dividend, 
minus  the  exponent  of  that  factor  in  the  divisor,  wh:n  the 
exponents  are  positive  fractions. 


216  ELEMENTARY  ALGEBRA. 

SIGHT      EXERCISE. 

Complete  the  following  expressions  : 

1.  G^i  =  6?"f^,  a^  =  a^^,  a^  =  «t^,  and  a^  =  ci^ 

2.  a'^  =  a^  —  «T^  =  ^2ir  —  a^T  =  cC^  =  aJ^  =  a^^ 

3.  «'i^  =  a^  ;  a^  =  a^;  a^^  =  a^  ;  a^-  =  a^ 

4.  a^Xa^=?   xi  xx^=?   x^  X  x^  X  x^  =  ?   x^  X  x^  ? 
6.a^.-^a^=?    c-^ci=?    c^i-^c^=?    c^  ^  c^  =  ? 
6.  a'^a'=?     (x^  Xx^)-^x^=?    {x^  -r-  x^)  X  x^  =  ? 


228.  We  learned  [P.  12,  91]  that  the  exponent  of  a 
factor  in  the  quotient  equals  the  exponent  of  the  same 
factor  in  the  dividend  minus  the  exponent  of  that  factor 
in  the  divisor.  If  this  principle  be  accepted  as  true  when 
the  exponent  of  the  divisor  exceeds  that  of  the  dividend, 

exponents  will  arise  from  its  application. 
Thus,  a^  -^  a^  =  a~^,  read  a,  exponent  minus  3  ;  also, 
-t-  a^  =  a~'^j  read  a,  exponent  minus  one  sixth. 

229.  «^'*  -^  a'""  =  a-^""  [228] ;  but 


^^— l-a^«[P.  53]  =  -^„ 


1 
a' 


a- 2"  in:  ^-  [Ax.  1].     Therefore, 

JPrin.  92. — A  quantity  affected  by  a  negative  exponent 
equals  the  reciprocal  of  the  quantity  affected  ly  a  numeri- 
cally equal  positive  exponent. 

230.  a''^  -^  a^""  =  a^^  [P.  12]  ;  but 

fl^2n_____  |-^j^^  -j^-j^     Therefore, 

Prin,  93, — A  quantity  affected  hy  a  positive  exponent 
equals  the  reciprocal  of  the  quantity  affected  hy  a  numeri- 
cally equal  negative  exponent. 


FRACTIONAL  AND  NEGATIVE  EXPONENTS.    217 

SIGHT      EXERCISE. 

1.  Find  the  value  in  negative  exponents  of  : 

ff*  -f-  a^  ;   a?  ^  x^  \   x  -^  x^ ;   a^  -7-  a^  ;   c^  -r-  c^ 

2.  Express  in  positive  exponents  : 

a-\  a-\  x-^,  x-K  2-^  S'S  {xy)-^   {^~' 

3.  Express  in  the  integral  form  : 

1     1     _L     J:_     i     _L 
a^'  X?'  2-2'  a;-^'  ar*'  x— 


231.  Since 

2      1  t. 

c-^d  ~  1  . ,  '-^-  ^'^-' -  ^  - ^-^  ^ rf  -:^' 

it  follows  that, 

JPrin,  94, — A  factor  may  be  transferred  from  either 
term  of  a  fraction  to  the  other  if  the  sign  of  its  exponent 
he  changed, 

232.  Since 

a-'  X  a-^  =  ^4  X  ^  [P.  92]  =  ^\  =  «"«  [P.  94],  and 

a-J  X  a-^  -  -\  X  \  [P.  92]  =  4i  =  «"**  t^-  ^^]»      ' 
.  a^      a*  ais^ 

it  follows  that, 

Prin,  95, — Tlie  exponent  of  a  factor  in  the  product 

equals  the  sum  of  the  exponents,  of  the  same  factor  in  the 

multiplicand  and  the  multiplier  when  the  exponents  are 

negative. 

233.  Since 

«-«  -^  a-^=  \.  --  \  [P.  921  =  i  X  f  =  ^3  =  «-' ;  and 
a'      a^  ^  ^      a^       i       a^ 

a    %^a    5  =  -.--  =  -.x^=^  =  a    *, 
at      flft      at       1       as 

it  follows  that, 


218  ELEMENTARY  ALGEBRA. 

Prin,  96, —  The  exponent  of  a  factor  in  the  quotient 
equals  the  exponent  of  the  same  ^  factor  in  the  dividend, 
minus  the  exponent  of  that  factor  in  the  divisor,  when  the 
exponents  are  negative. 

SIGHT      EXERCISE. 

1.  Clear  of  negative  exponents  : 


a-^      a^      a-'^     a'^l     x'"^     xy-'^     a-^h-^ 

a^h 

J-3.     ^-2'       ^,2     .     ^-2^^    ^_|.        ^-1     .              2           ' 

c-^ 

2. 

Express  in  the  integral  form  : 

a     a~^      a      Sa^         ax         x-"^        mn^ 

I'     J2  ?    j3?    j-4?    5-1^-1'    2^-4'    p-^q-^ 

3. 
4. 

Express  in  the  integral  form  : 

a     «2    ^-2    an    ah        1          2        2 
c'  r^'  x-^'  c-^'  cd'  a-'P'  a-''  2"^ 

Express  in  positive  exponents  : 

a-^    a-^W    x-^if    2-^x3     {a-\-h)-^ 
j-2>     ^-3  ^       ^2     .       5-1     ^  {a-\-h)-' 

5. 

Find  the  numerical  value  of  : 

6.  Find  the  value  of  : 

a^  X  a-^ ;   a^  X  a~^ ;   x-^  X  a:~^  ;   a-^  X  «~*  ; 
«-'X«-*;   2;-^X:r-^;    S-^xS-^;    4-2x4^ 

7.  Find  the  value  of  : 

x^-^x-'^',  a~^-^a~^',  a~^  ^  a~^ ;  a~^ -^  a~^  ; 

a-^  ^  a-^ ;  x-^  -^  x'^  ;  2"*  -^  2-« ;  2"^  -^  2"^ 

8.  Find  the  value  in  positive  exponents  of  : 

a-^  X  a-^ ;   a"^  -^  «-^ ;   a~^  X  a^  ;   a~^  -^  a^ 
a^      x~^y^     (a  —  h)~'^     {x -\- y)^      p~^  q^ 


FRACTIONAL  AND  NEGATIVE  EXPONENTS.    219 

2.   General    Principles. 

234.  1.  a'"  =  aXaXaX to  m  factors. 

a''=aXaXaX to  n  factors. 

. • .  arxar={aXaXa to  m  factors)  X(aXaXa 

to  n  factors)  =  axaXa to  (7n-{-n)  factors  =  «"*+* 

2.  af  =  /A  [P.  89]  =  ('Vfl)^-  [224] 
flT  =  al^  [P.  89]  =  (V^)-^  [224] 

.-.   af  X  a*  =  (Vay  X  CVa)-^  =  ('V^)'-+"^  [1]  = 

a    «»     [P.  75]  =  a  9  ^  « 

3.  a-^X.:"  =  ^xi[P.92]=-i^.[l]  = 

«-<•»+'•>  [P.  94] 

4.  a-f  xa-^  =  -^X-^  =  -^^^  [2]  = 

-1^)  [P.  94]  =a-(f  +  T) 
Therefore, 

JPrin.  97*  (f  X  a*  =  a'^^  for  any  positive  or  negative, 
integral  or  fractional,  values  of  x  and  y. 

235.  a'^-^a*^  a""-*,  since  a"""  X  a"  =  a"  [P.  97] 

p^  r_  p f_  £__!_  -L  JL 

ai  -^  a*  =a<i     »,  since  ai     »  x  a*  =  a^  [P.  97] 
«-"*  -^  «-"  =  a*-"*,  since  a""'"  X  a~*  =  «""*  [P.  97] 

_p^  _^  r p_  L.  —  L.  _JL  —L. 

a    9  -^  a    *  =a»     ?,  since  a*     t  X  a    *  =a    ? 
[P.  97].     Therefore, 

Prin,  98.  a"  -r-  a'  =  a'~^  for  any  positive  or  negative, 
integral  or  fractional,  values  of  x  and  y. 

Scholiuni, — Principles  97  and  98  are  stated  in  general 
language,  although  the  cases  in  which  one  of  the  quantities 
considered  has  a  positive  and  the  other  a  negative  exponent 
are  omitted  in  the  demonstrations.  Let  the  pupil  supply 
the  omissions. 


220  ELEMENTARY  ALGEBRA, 

236.  If  we  let  x  be  general,  and  let  y  =  n,  — ,  —n, 

p  ^ 

and  —  —  successively,  then  will 

1.  («'')«  =  ^*  Xa'Xa' to  ^  factors  =  «'*  [P.  97] 

2.  (a^)  i  =  V(^  [223]  =  V^ [1]  =  af  [P.  75]  =  a' " 7 

3.  («r" = (^  [P.  m = ^»  [1]  =  «-*"  [P.  94] = «-<-«) 

4.   («^')-f  =  -i^  [P.  92]  =  4  PI  =  ^"'^  [P*  ^^]  = 

Therefore,  a'H-y; 

Prin,  99<,     {a^y  =  «'«'  /or  any  positive  or  negative,  in- 
tegral or  fractional,  values  of  x  and  y, 

237.  1.  {ahY=:ahxahXah to  n  factors  = 

(aXaX  a to  n  factors)  X  {hxhxl) 

to  n  factors)  =  «"  X  ^^ 

2.  (a b)!-  =  V{aby  [223]  =  V^^V^  [1]  =al-xh'<f  [P.  76] 

4.   (a^)-f  =  -i-  [P.  92]  =-^^  [2]  = 
(a5)?  ai  X  hi 

Therefore,  a~f  X  ^~t  [P.  94] 

Prin,  100.    {a  b)'  and  a'  X  b'  are  equivalent  for  any 

positive  or  negative,  integral  or  fractional,  values  of  x. 

.     238.  If  we  let  x  be  general,  as  in  the  preceding  articles, 
we  have, 

(~X=  (a  b-'Y  [P.  94]  =  ^^  X  b-^  [P.  100]  =  f^  [P.  94]. 

Therefore, 

/aY  a' 

Prin,  101,    [j\  and  j,  are  equivalent  for  any  posi- 


ive  or  negative,  integral  or  fractional,  values  of  x. 


MISCELLANEOUS  EXAMPLES.  221 

Miscellaneous  Examples. 

EXERCISE     111. 

1.  Find  the  value  of  8^  16*,  27^,  and  32^ 

2.  Find  the  value  of  (-  64)^,  (-  125)^,  and  (-32)^ 

3.  Find  the  value  of  (27)-*,  (625)"*,  and  (-  64)-^ 

4.  Find  the  value  of  4^  X  4^,  2^  X  8*,  and  3*  X  9* 

5.  Find  the  value  of  (2  X  8)^,  (3  X  9)*,  (4  X  8)^  and 
(2  X  2  X  2)* 

6.  Find  the  value  of  (4  X  16  X  25)^  (8  X  27  X  64)* 

7.  Simplify  {x-^)^  ]  x-^  x  x^  -,  x-^  X  {aa?)^ 

8.  Divide  a^x~^  by  ax^ -,  x~^y^  by  x^  y^ 

9.  Express   in   positive  exponents   ax~^;    a~^b~^c*; 
and  (x-^)-^ 

cij*  x^  ij         X    "  ■?/*  j2~"* 

10.  Reduce  to  lowest  terms  :  £-^  ;   — j-^ — r- 

a~^x^y^       x^y~^z^ 

11.  Expand  («^  + 5^)2;   («i  _  ^,i)2 ;   («i  4.  ^,i)  (^i  -  Z,^) 

12.  Expand  (x*  +  ?/*)2;  {x^-y^^-^  («^  + j-|)(rtf  _  j-f) 

13.  Expand  (a:* +  y*)^;   {x-' -  y-^f -,    /^-s  +  AV 

14.  Multiply  x^  -\-  x^  y^  -\-  y^  by  x^  —  y^ 

15.  Multiply  x~^  —  y~^ -^  x~'^  y~'^  by  x^-^-y^ 

16.  Multiply  x^  —  x^  y^  -{■  x^  ij^  —  ^y^-\-  y^  by  x^  -j-  y^ 

17.  Divide  x^  —  y^  by  x^—y^;  x~^  —  y~*  by  x~^-\-y~* 

18.  Divide  x^ -^  x -\- x^ -{- 1  by  a;*  +  2  a;*  +  1 

19.  Multiply  (a«+a:2)8  ^y  (a^^a^)^;  (x"-}-!)^  by  (s^-1)^ 

20.  Multiply  {x" -\- X -{- 1)^  by  {a^-x-{-l)^; 

x~^-\-y~^  by  x~^-{-y~^ 

21.  Resolve  into  two  factors  a  —  b  ;  x -{- 2  x^  y^  -\- y 


222  ELEMENTARY  ALGEBRA. 

22.  Factor  a^  —  h^;   a*  +  J*  ;   a^  —  h^ 

23.  Expand  {a^  +  l^f  ;   {a-^  -  J-^)*  ;   (a^  -  «-t)5 

24.  Express  in  simplest  form  with  positive  exponents  : 


Radicals. 


Definitions  and   Principles. 

).  Any  quantity  affected  by  the  radical  sign  or  a  frac- 
tional exponent  is  a  radical ;  as,  V^,   Va,  A 

240.  A  quantity  without  a  root  sign,  or  one  whose  indi- 
cated root  can  be  exactly  obtained,  is  a  rational  quantity, 

241.  A  radical  that  can  not  be  reduced  to  a  rational  quan- 
tity is  an  irrational  radical,  or  a  surd;  as,  V3,   Vb,   Va. 

242.  A  surd  that  expresses  an  eyen  root  of  a  negative 
quantity  is  an  imaginary  surd,  or  imaginary  quantity; 
as,   V— 5,   V  —  a^, 

243.  Any  quantity  that  is  not  imaginary  is  a  real  quan- 
tity ;  as,  5,  —a,   ±  Va^,   ±  Vb. 

244.  A  factor  placed  before  a  radical  to  show  how  many 
times  it  is  taken  is  the  coefficient  of  the  radical. 

246.  A  radical  that  has  no  factor  whose  indicated  root 
may  be  found  is  a  pure  radical;  as,  VH  ab. 

246.  A  radical  that  has  one  or  more  factors  whose 
indicated  root  may  be  found  is  a  mixed  radical;  as, 
VI^{=  ±2a\/b). 

247.  The  degree  of  a  radical  is  denoted  by  the  index  of 
the  indicated  root.     Thus,  a/5  is  of  the  third  degree. 


RADICALS.  223 

248.  A  radical  is  in  its  simplest  form  when  it  is  pure 
and  is  in  the  lowest  degree  to  which  it  can  be  reduced. 

249.  Radicals  are  similar  if  they  contain  the  same  surd 
factor  when  they  are  made  pure.  Thus,  V4  a^  by  which 
equals  ±2  a  Vb,  is  similar  to  3  Va^b,  which  equals  ^aVb, 

h  JL        i. 

250.  Since  {ab)*  and  ««  X  ^*  are  equivalent  [P.  100], 

it  follows  that  VoT  and  Va  X  Vb  are  equivalent. 

Therefore, 

Prin.  102. — Any  root  of  the  product  of  two  quantities 
equals  the  product  of  the  like  roots  of  those  quantities. 

Prin,  103. —  The  product  of  the  equal  roots  of  two 
quantities  equals  the  like  root  of  their  product. 

a\ »        -,    a* 


251.  Since    l-r)      and  -y  are  equivalent  [P.  101],  it 


follows  that   i  /  y  and  ■;^  are  equivalent.     Therefore, 

Prin,  104, — Any  root  of  the  quotient  of  two  quantities 
equals  the  quotient  of  the  like  roots  of  those  quantities. 

Prin.  105* — The  quotient  of  the  equal  roots  of  two 
quantities  equals  the  like  root  of  their  quotient. 


^ *^  =  i  Va¥^,  it  follows  that, 
b  0 

Prin.  106. — N^o  fractional  radical  is  pure. 

253.  Since  a  =  Va",  it  follows  that, 

Prin.  107> — Any  quantity  equals  the  nth  root  of  the 

nth  power  of  the  quantity. 

JL  ±         1 
264.  Since  («•)«•  =a— i  [P.  99],  we  have, 

Prin.  108. — The  V\/~a  and  the  "Va  are  equivalent. 


224  ELE3IENTARY  ALGEBRA, 

Reduction  of  Radicals. 

Problems. 
1.  Mixed  to  pure  radicals, 
ninstrationsc — 

Keduce  Vs a'^ b  and    a/  —  to  pure  radicals. 
Solutions : 


1.  ^San  =  V4«^x  2ab=  \/Aa^  x  y\/2ab  [P.  102]  = 

±2a\/2ab. 

2-    r    2^  =  4/2^  Xy=:  4/125-  =  y  125-  ><   15  =  ^125-  >^    V15  ^ 

£P.  102]  =  I  Vl5. 

EXERCISE     112. 

Eeduce  to  pure  radicals  : 

1.  V12,   a/is,  and  \/33  3-  VT6,   V-54,  and  v^i28 

2.  V45,  Vis,  and  a/75  4.  ^-56,  Vi08,  and  a/ISS 


5.  \/4^^  Vl6^^  and  V^c* 


6.  vieT^  V18^,  and  V-8«* 

7.  VsOfl^,   a/45^^,  and  V27^V 


8.  VTe^^,   V-:?;^«/2;S  and  V^V^ 


9.  V«2(«  +  ^')  and  Va(a+^ 


10.  '\/x{x-\-yY  and   a/:z;  (:ic  +  1^)^ 


11.  V (x -\- yY  {x  —  yf  and  Va;^  5/^  (a;  +  yY 

12.  Va^  +  2  rz;^  y  +  a;  «/^  and  Vx  {x  -\-yY 

13.  VH^-y'){x^y) 

14.  Vfl2Z>-^(a=^-Z'^)(a-^) 


REDUCTION  OF  RADICALS, 


225 


17 


18. 


19. 


8      3  /16         1    3  /27 

9'  1/27'  ^'^  l/re 


G^% 


x^,  and 


a-\-b 


24. 


26. 


X 


4/^^  ^"'  1/^ 


y)' 


"•i/pf-'i/lxf 


2.  To  lower  degree. 


niiistratioiis.- 


Reduce  VOo^  and  V64a^  to  simplest  form. 


Solution :   1.  V9a«  =  VV^a*  [P.  108]  =  V^oT. 


2.  V64o»  =  vVWo*  [P.  108]  =  v^=  \/4x2a  = 

V^x  \/2a=  ±  2 


226  ELEMENTARY  ALGEBRA, 

EXERCISE     118. 

Eeduce  to  simplest  form  : 
1.  V9,   V64,  and   Wl  7.  Vsi,   Vl25,  and  V729 

2  iA  iA  i/^      8  iA  ]/^  ^//"^ 

3.  Vl6^V  and  V25^V        9-  V^V^and  V-^x^y'' 

4.  V^V^  and   VcF^'^      10.  V^3_3^2^_|_3^j2_j3 

^-  y  ^^ ^^^  y -.:?     "■  y -27?  ^"^  y -^ 

6.  V-32a5/«  and  V^V^         12.  Va2  +  2a^  +  Z>2 

3.   Bational  to  radical  quantities. 
Illustrations. — 

1.  Reduce  2  a  to  a  radical  of  the  third  degree. 
Solution:  2a  =  V(2a)3  [P.  107]  =  VSo^ 

2.  Free  %aXf%~a  of  its  coefficient. 
Solution : 

2a  V2a=  V(2a)3  x  XJ%az=.  \f^  x  ^^0^=  Vl^a^  [P.  103]. 

EXERCISE     1 14. 

2 

1.  Reduce  5,  3  x,  and  —  o;^  to  radicals  of  the  second 

degree. 

2.  Reduce  3«,  5  5a7,  and  -  to  radicals  of  the  third 

X 

degree. 

3.  Reduce  a^lP',  xy^,  and  -^  to  radicals  of  the  fifth 
degree. 

4.  Free  2^5,  3  a/3,  and  -  Vs  of  coefficients. 

5.  Free  a  Vd~a,  7?  Vd~Xy  and  -  V?  of  coefficients. 


REDUCTION  OF  RADICALS,  227 


6.  Free  a(a  +  h)^  and  {a—h)  Va  +  b  of  coefficients. 

7.  Reduce  a?,  y^,  and  4  to  equivalent  expressions  having 
an  exponent  of  - 


8.  Free  x{y)^  and  a^{z)^  of  coefficients. 

9.  Free  a;^  {yY  and  16  (2:)*  of  coefficient 

(2\l 
1  +  l^j  ={x'^y^)i 


4.  Keduction  to  same  degree. 
Hlxistration, — 

Reduce  Vs,  V2,  and  a/2  to  the  same  degree. 

Solution :      ^3  =  3^  =  3^  [P.  89J  =  'V3«  =  ^l/m 
V2  =  3*  =  2t^  [P.  89]  =  V2"'  =  'VI6 
V 2  =  2i  =  2^i  [P.  89]  =  V23  =  *y8 

Kote. — The  operation  may  be  shortened  by  remembering  that  both 
index  and  exponent  may  be  multiplied  by  the  same  number  [P.  89]. 

Reduce  Vc^,  Vc^,  and  Va*  to  the  same  degree. 

Solution  :  l/d^  =  'V(^  [P.  89]  =  V^* 
V^5  ^  ly  (^«  [-p,  89]  =  Va^" 
^~a*  =  V^  [P.  89]  =  V^ 

Note. — The  common  index  is  the  L.  C.  M.  of  the  given  indices. 
Why! 

EXERCISE     lis. 

Reduce  to  the  same  degree  : 

.  VI  VI  ^  VI  ,^,^.„,|/T 

2.  Vx^y   vary,  and  Vy^  ./—  e^ 

6.  a,   va^  and  Va^ 

3.  aK  hK  and  c^  ^    ^'^  y^^  ^^^  jy^ 

4-  Va-\-h  and   l/a-{-  b  8.  Var  (a;  -|-  y)  and  (a:  -f-  y)^ 


228 


ELEMENTARY  ALGEBRA. 


Addition  and  Subtraction  of  Radicals. 

Illustrations. — 


1.  Find  the  sum  of  4  \/8,  5  i/i,  and  |  a/- 
Solution  :  4  \/8  =  4  V4  x  3  =  4  x  V^  x  ^2^=     8  V^^ 

|t/|  =  |4T^=|  X  4/1  X  V2=    i-V2 


2.  Find  the  value  of  2  V-  81  a*+ 8  Vs^-  2  «  y-  24 «. 
Solution : 

2  V-  81  a*  =2v^-27a3x3a  =  3x  V-27a3x  V3^=-6aV3^ 
+  8V^  =  +  8Va^x  3a  = +  8  X  V«^  X  VSa  =  +  8aV3a 
-2a  V-24a  =  -2a  V-8x3a  =  -2ax  V^x  V^=  +  4aV3a 

Sum  =     6a  v^ 


EXERCISE     116. 


v?+i/^+v? 


Find  the  value  of  : 

1.  Via -^  Vl6a -\- VS6a 

2.  a/2  +  \/8  -  iA  , 

/-  ^  /—  ^V"-^W^^^ 

3.  V27x+Vl2x-V4.Sx  \    a-b       b 

3  / 3/5  1    3  y ,  ^       3    /o^  3    /o^     ,       3    /  J 

5.  \/2^+  \/2F+  ^2^        11.  V7«^+  VY^-  a/7? 

6.  'v/(a+T)^+V'(^^^^^     12.  V«^2_|_«/^2_  3/7^2 

7.  a/^- V^+a/^         13.  2^/3+ V27  +  3  Vsi 

15.  (a^ ^^a^h^aW)^ -{a^-2 aH-\-ai^)^ 

16.  Va^  -2aH-^ab^  ±  Va^ +  2aH -\- ab^ 


MULTIPLICATION  OF  RADICALS.  229 

19.  -i-^  a/?+^+  i/— ^ 
a-b         ^       ^  y  a  +  h 

20.  A/  -^L—^xVax  —  x^ 
\  a  —  x 


17 


18 


Multiplication  of  Radicals. 
Illustrations. — 

Multiply  3  yi  by  5  i/|,  and  3  \/2  by  2  Vs. 
Solutions : 

1.  5l/|x34/|  =  5x3x/|xl/I=:15  /f^  [P.  103] 

=  15/|  =  15i/|  =  15>/|x  VS" [P.  102]  =5^/3: 

2.  2  V3  X  3  a/2  =  2  X  3  X  V3  X  y^  =  6  x  {/O  x  Vs  [P.  89] 

=  6^72  [P.  103]. 
EXERCISE     117. 

Find  the  value  of  : 

1.  V3  X  a/6  7.  A/3rt"  X  vTrt  X  vT« 

2.  V3X  V6  '  Z.'^txViy.ZyV'iy.zV'x 

4.-4  a/2^  X  3  Vac  lo    i  /— £—  X  i  A  +  ^ 

/-  -  V  a  +  b^V  a-b 

^•yfxy*  u.  a/5"xV2 


c 


l/i^Vl^^ 


12.  2A/3X3  V3X  V3 

13.  2  Vfl^  X  —  3  Va^ 


230 


ELEMENTARY  ALGEBRA, 


14.  (a/2  +  V3  +  Vb)  X  a/2 

15.  {Va  +  Vh){Va-Vl) 

16.  a/o+T X  ^/a  —  b  X  A/a^  + 


«  +  ^       3  /a  —X  _  3/7 ^ 

18.  a  ^/xy^  X  5  V?p  X  c  V^^ 

19.  (3  a/2  -  4  a/6  +  5  a/10)  X  a/6 

20.  (2  a/^-  3  Vy)  (2  a/^+  3  a/^) 

21.  (ic  -  a/^+  y)  {Vx-\-  Vy) 

22.  (a;  +  a/^  +  ^)  (^  -  V^+  ?/) 


Division  of  Radicals. 
Illustrations.— 

Divide  3  j/l  by  2  l/|,  and  V%  by  Vs. 
Solutions : 

1^  =  1^1  X  Vl2  [P.  103]  =-1^12. 

2.  V2-5-  V3=  V8-5-  V9=  V8T9  [P.  105]  =^  = 

V'^^^STI^  =  t^"^  =  ^^  =  -  Vl28. 


EXERCISE     118. 


Find  the  value  of  : 

1.  a/8-4- a/2 

2.  4  a/is  -4-  2  a/3 


4.  5  a/30  -t-  3  a/6 

5.  Va/b^  Wc 


DIVISION  OF  RADICALS.  231 

l.'UVa^-^Vab  13.  6\/3  x2a/5-^  Vl5 


14.  (2  a/30  -^  -  \/2)  X  V6 
9.  V2  ^  V2  ,- 

10.2-^V2  15.aV^X^A/^^|^| 

12.  2\/3x3\/2-v- 4/1  ,^- /-     ^ 

18.  (a/6  +  V^+  VTO)  -J^  a/2 

19.  (3A/lO  +  4A/5-6Vi5)-T- V5 

20.  (5 a VaJ^  —  b^  —  )i0ab's/a-\-b)^6aVa-\-h 

21.  (iC  +  2  A/a^y  +  i/) -j- (V^+ \/^) 

22.  (o^^xy-^r  y^)  -^  (a^  +  a/^+  y) 


Involution  of  Radicals. 

niustrations. — 1.  Raise  v5  to  the  second  power. 
Solution  :  ( V^)'  =  (Si)'  =  5l  [P.  99]  =  V^*  =  \/25. 

2.  Raise  Va  to  the  third  power. 

Solution :  ( V^)«  =  (ai)»  =  at  [P.  99]  =  ak  [P.  89]  =  ^/~a. 

3.  Raise  Va  to  the  sixth  power. 

Solution :  ( V")"  =  iahf  =  aS  [P.  99]  =  a^  [P.  89]. 

EXERCISE     1 19. 

Find  the  value  of  : 

1.  {V%f  6.  (2  V5)«  '  9.  {{a-\-h)VahY 

2.  (a/3)»  6.  (a  Vabf  lo.  (A/a  -  Z')=* 

3.  (2a/2)«  7.  (a/2^)*  11.  (a/^T^)* 

4.  (3  V^f  8.  (V^)«  12.  (V^^^f 


232                      ELEMENTS 

IRY  ALGEBRA. 

"•  (i/l)* 

-m' 

14.  {V~a+Vhf 

20.  (2  +  V3)' 

...(.vi)- 

"•(/i-/D" 

16.  {Vx  —  Vyf 

22.  (Vs-iy 

"  m 

23.  (^lA/2  +  iA/3)^ 

18.  (2  a/3  -  3  V2f 

24.    {-«(V^-VI)l 

Evolution  of  Radicals. 

Illustrations. — 1.  Extract  the  square  root  of  Vs. 
Solution :  i^l/S  =  V  5  [P.  108]. 

2.  Extract  the  cube  root  of  Vs. 

Solution :  \/^8=  V^  [P.  108]  =  VJ/S  [P.  108]  =  V2. 

3.  Extract  the  square  root  of  8  V2. 
Solution :  \^^i/2  =  |/4x2  V^  =  VT^H/W  = 

V4  X  i^J/TS  =  ±  2  V^ 

EXERCISE    120.  ^ 

Find  the  yalue  of  : 
1.  VVSa  6.  \/J/7W7 


2.  /W?  7.  V2V2 


3.  4/2  a  V2  a  8.  ^  -  Vs 


EVOLUTION  OF  RADICALS.  233 

11.  Vl6V2a  18.  i^27V2aa^ 

12.  VV(a  -  by 


19. 


Vi-\/l 


13.  VV?  +  27h^ 

14.  ^7(^T^^  20.  \/A:i  V{a-\-b) 

15.  V2Vl5(a  +  a:) 

16.  V'A/(a  +  J)(a-J) 


21. 


ViVi 


17.  yA/a3  +  3«2J  +  3aZ>2^^,3 


Rationalization. 


255.  To  rationalize  a  quantity  is  to  clear  it  of  radicals. 

2 
niustrations. — 1.  nationalize  the  denominator  of  — r= 

V3 

Solution :  —^  =  —p- — ^^-r^  [P.  56]  =      :.     =  tt  V  3. 

2.  Rationalize  the  denominator  of  — ■]= 

2-^/3 
Solution : 

2+V3^(2+\/3)(2+\/3')^4  +  4V3+3^^     ^^ 

2_/v/3      (2- V 3) (2+ a/3)  4-3  ^ 

EXERCISE     121. 

Rationalize  the  denominators  of  : 

\  a  2 


1. 


VT  a/F  2+V2 

„      2  a/7  3 


3. 


V5  ^- v^»  *^-V2-vT 

a/3"  1  g 

a/5"  ^'  1  -  a/ 2"  ^'  a  -  VT 


234 


10. 


11. 


ELEMENTARY  ALGEBRA. 

3- V2" 


Vx  +  Vy 
2-^2" 


12. 


13. 


3  +  V2 

V2- V3" 
a/2 +V3~ 


14. 


15. 


Vic  —  a/^ 


16.  If  the  a/2  =  1-4142,  what  is  the  value  of  -7^  ^ 

17.  What  is  the  numerical  value  of  7=  ? 

2  +  A/2 


Imaginary  Quantities. 

.   Principle  and   Definition. 


256. 


_  4  =  a/4  X  (-  1)  =  a/4  X  a/^  [P.  102]  = 


±2a/-T. 

a/^=^=  a/6  X  (-  1)  =  a/6  X  a/-1  [P.  1021  = 
Therefore,  ±  a/6  (a/^^). 

JPrin,  109, — Every  imaginary  quantity  of  the  second 
degree  may  he  reduced  to  the  form  of  ±  x  ^J  —  1,  in  which 
X  may  he  rational  or  irrational. 

257.  The  factor  a/—  1  is  the  imaginary  unit.  —  a/—  1 
is  equivalent  to  —  1  X  A^—  1. 


2.  Examples. 
EXERCISE   122. 
Keduce  to  simple  form  : 


1.  V— 9,    a/— 4a^   and  V— 16 


2.  a/- 25:^2,    V- 36^2  2:*,    and  A/-49a*2/^ 

3.  a/- 8,    a/-  12  «,    and  a/- 18  «2 0^3 

4.  Add  a/^^,    V^-^,    and  a/- 16 


IMAGINARY  QUANTITIES.  235 

6.  Find  the  value  of  V-^a""  +  V-25a^  -  Vl6^' 

6.  Simplify  {V^y,  (v'^l)^  (V^i)S  (V^iy 

'  7.  Simplify  (-  V^^)%   (-  ^/^)^    (-  V^=T)S 

and  (-  V^^Y 

8.  Multiply  V^^  by  \/^ 

Suggestion.  V— ^  =  V^  x  ^/^^  and  v^^  =  ^/^  x  ^  —1; 
hence,  V— ^  x  ^/^  =  aA  x  v^  x  V—  1  x  a/— ^  =  V^^  x 
(Vin;)«  =  Vl8  X  (-  1)  =  -  Vl8  =  -  3  V2. 

Multiply : 

9.  V^^  by  V^^;    V^^by  V-20; 

10.  2  \^^  by  3  V"^^;  4  V- 16  by  2  \/-25; 

a  V-b^  by  5  V^^ 

11.  2  +  V^^  by  2  -  V^;   V'^^H-  \/^^  by 

12.  Square  2  +  v^^^;  3  +  V^^;   V^-  V^^ 

13.  Divide  V— 36  by  V^^ 

Suggestion.    V— 36=  V^^  x  —  1  =  6  V—  1 ;  a^d  V— 4  = 

V- 4      2V-1      ^ 
Divide : 

14-  \/^^  by  V^^;    V-12  by  a/^^; 

^Tby  v^^ 

15.  2  V-4a:2  |5y  V-a;^;  V-16a:*  by  V-2a:; 

Rationalize  the  denominators  in  : 


^/^^by  V^ 


236  ELEMENTARY  ALGEBRA. 

Square  Root  of  Binomial  Surds. 

I.   Definitions  and   Principles. 

258.  A  binomial  one  or  both  of  whose  terms  are  surds 

is  a  Mnomial  surd ;  as,  a  ±  ^/h,  Va  ±  V^. 

Note. — The  discussion  in  this  section  will  be  limited  to  binomial 
surds  of  the  second  degree. 

269.  Since  the  rational  term,  if  there  be  any,  may  be 
put  in  the  form  of  a  radical,  and  the  coefficients  of  the 
terms,  if  there  be  any,  be  placed  under  the  radical  sign — 

JPHn,  110. — Every  Unomidl  surd  of  the  second  degree 
may  he  reduced  to  the  form  of  V^ ±  vh,  in  which  one  of 
the  terms  may  he  rational. 

260.  The  square  of  {\^  ±  Vh),  or  {Va  ±  Vhf  = 
«  ±  2  VoT  -\-h  =  {a-\-h)  ±2  Va/b,  ■  a  binomial  surd. 

Therefore, 

Frin.  Ill, — A  binomial  surd  may  he  a  perfect  square, 
and,  when  it  is  the  square  of  a  binomial  surd  of  the  second 
degree,  one  of  the  terms  is  rational, 

261.  Since 

{Va  ±  ^fhY  =.(a-^h)±%  Vah,  {a-\-b)±2  V^  is 
the  type  of  a  binomial  surd  that  is  a  perfect  square. 

Therefore, 

Prin,  112. — A  binomial  surd  with  a  rational  term, 
and  the  coefficient  of  the  irrational  term  reduced  to  ±  2, 
is  a  perfect  square  when  the  quantity  under  the  radical 
sign  is  composed  of  two  factors  whose  sum  equals  the  ra- 
tional term ;  and  its  square  root  equals  the  sum  or  differ- 
ence of  the  square  roots  of  these  factors. 

262.  {a-{-b)  ±%  Vab  is  the  type  of  a  square  binomial 
surd.  Now,  {a-\-bY-{±%Vahf=a^-{-^ah-^l^- 
4tab  =  a^  -  2ab  -\-  b^  =  {a  -  bf.     Therefore, 


SQUARE  ROOT  OF  BINOMIAL  SURDS,  237 

Brin,  113, —  When  a  Unomial  surd  is  a  perfect  square, 
the  difference  of  the  squares  of  its  terms  is  a  perfect  square, 
and  is  equal  to  the  square  of  the  difference  of  the  two  factors 
described  in  Frin.  112. 

2.  Examples. 

Ulnstrations. — 1.  Extract  the  square  root  of  14  +  6  V^. 

Solution  :  14  +  6  ^/~E=  14  +  2  y^.  The  two  factors  of  45  whose 
sura  is  14  are  9  and  5 ;  therefore,  the  square  root  of  14  +  2  ^45  is 
±  ( V^  +  V5)  [P.  112]  =  ±  (3  +  VS). 


2.  Extract  the  square  root  of  2  a  —  2  Va^  —  t^. 

Solution :  The  two  factors  of  a'  —  &*  whose  sum  is  2  a  are  a  +  6 


or 


and  a  —  h\   therefore,  y  2 a  —  2  Va*  —  6'  =  ^Ja  +  &—  ^a  —  h 
^Ja  —  h  —  ^Ja  +  6  =  ±  {^/a  +  ft  —  ^/a  —  h). 

3.  Extract  the  square  root  of  81  —  36  Vb. 

Solution :  4/8I  -  36  //S  =  \/^\  -  2  Vl620.  The  two  factors  of 
1620,  whose  sura  is  81,  are  not  readily  seen.  Let  x  equal  one  and  y 
equal  the  other.    Then, 

(1)  a;  +  y  =  81 ;  and 

(2)  (X  -  yf  =  81«  -  (2  VT620)«  =  81  [P.  113J 
.-.    (3)       x-y  =  9 

Add  (3)  to  (1)  and  subtract  (3)  frora  (1), 
2a;  =  90  and  2y  =  72 
x  =  45  and     3/ =  36 

•••    4/8I  -  2  V1620  =  V36  -  V45»  or  V^ -  V^  = 

6  -  3  -/S  or  3  ^/5^  6  =  ±  (6  -  3  v^) 

EXERCISE     128. 

Extract  the  square  root,  when  possible,  of  the  following 
expressions  : 

1.  5  +  2\/6  3.  7-I-4A/3  5.  15  +  6  a/6 

2.  9  -  4  a/5  ,  4.  8  -  2  vis  6.  4  +  a/3 

7.  3a-2a\/2  9.  11  +  2a/30 

8.  6a;  +  2a;A/5  10.  13  —  2  a/42 


238 


ELEMENTARY  ALGEBRA. 


11.  16-4\/20 

12.  2a;  +  2  Va;2  — ^2 

13.  15-2  V56 

14.  2a;  +  2  Va;^-4/ 

15.  22-4  VSO 

16.  (a;  +  ?/)  —  2  V^ 


'••5  +  3^ 


23.  15  -  V56 

24.  («  +  1)  +  A/4a 

25.  ^i  +  ^4/l| 

26.  1  -  I  V6 

o 

27.  25  +  2  \/i56 

28.  -3  +  4\/^ 


19.  (l  +  2a;)  +  2Va:2  +  a; 

20. 9  +  Vn 

21.  23  -  ^528 

22.  16  +  Vise 

35.  (x -{-yY •—  4t{x  —  y)  \fxy 


29.  x  —  y  —  %v  —  xy 

30.  42  +  36  a/2 

31.  41-4  Vl05 

32.  0  +  2  a/^ 

33.  6  +  V35 


34.  10- V 100-4  a;2 


36.  (2a;  +  l)  +  2Va;2  +  a;-2 


Miscellaneous  Examples. 

EXERCISE     124. 

Express  with  fractional  exponents  : 

1.  V^ 


3.  Va^^l* 


2.  V{aH(^Y  4.  V(^  +  Z')2 


5.  l/S{x-yf 


"v>^ 


'V(a»  -  h^Y 


7.  V(a2  -  ^^)"  8. 

Express  with  the  radical  sign  : 
9.  x^  11.  a^h^        13.  {a-{-x)^        15.  a;^(a;  +  2^)* 

10.  {a  c)^      12.  a;^  y^       14.  (a^  -  a;^)^      16.  a^  {a  +  J)^ 


MISCELLANEOUS  EXAMPLES,  239 

Express  as  mixed  surds  : 

17.  3V3  20.^3  3/^  22,  xyV^Ty 

lQ.a'V¥  21.  a:tV^  23.  |  VS 


19.  {a  +  b)  Va-\-b  24.  {a  +  a:)  Va  —  x 

Place  the  coefficients  of  the  following  expressions  within 
the  parentheses  : 

25.  3  (3)^  29.  7?  (a.'2)-8  33.  «-^  (a"')-' 

26.  4  (2)^  30.  2^  («^-2)-*  34.  a;^  («  +  J  a;-^)* 

27.  4  (4)^  31.  -  8  (a  -  5)^  35.  x'^  (a  +  a;)"* 

28.  a  (of  32.  a  (a-3  J)i  gg.  ^f  (^-f  )^ 

Reduce  the  following  expressions  to  equivalent  ones 
having  a  coefficient  of  2  : 

37.  6  a/3  40.  7  V2  43.   -  8  XT^ 

38.  5  Vo"  41.  -  3  V4  44.  8  (a  -  If 

39.  V20  42.  3  (a  -  h)^  45.  54  (a:  +  yf 
Complete  the  following  expressions  : 

46.  (8aa;)^  =  4(    )^  hO,  (a^ 7?)^  =  w' {    )* 

47.  (16  a;  2^)^  =  8  (     )^  51.  (a;^  y^)\  =  ^^l  (    )f 

48.  (32a#  =  4(    )^  52.  av(a-i)^  =  (    )^ 

Express  as  pure  surds  : 

54.  Va3  +  2«2^,  +  aZ^  66.  """VS^ 


58, 


65.  V(a  -  x)  {a^  -  a:^)  57.  Vax^^* 


240 


ELEMENTARY  ALGEBRA. 


Simplify  : 

62.  7  V54  +  3  Vl6  +  a/432 

63.  {x^y^-^ay^)^-2{x'z^-^az'f 


64.  V(a  +  Z>)-^X  a/(«  +  ^)~' 

66.  ( VU  +  A/2i  -  V42)  ^  V7 

67.  ( V40  -  Vl6  +  V56)  -s-  VS 

68.  {a  —  x)-^  ( V«  —  V^) 

4  V40      14  vT2     2  Veo 


69. 


3  Vl08        5  Vl4  *   3  V84 


Reduce  to  equiyalent  forms  having  a  rational  denom- 
inator : 


70. 


71. 


^x  V6a 
Vda 


V2 
VI 


73. 


74. 


75. 


a-\-Vb 

4 

V3  -  V2 

2 


76. 


77. 


78. 


3  +  2  V2 
V5"-  Vs 

1-  a/^^ 
1+ V^^ 


V-3+  V-5 


1-  V5 
Arrange  in  the  increasing  order  of  magnitude  : 

79.  3  V2,  2V3,  Vis  .       81.  Ve,  a/15,  2  V3 

80.  a/5,    a/IO,    2  a/2  82.  a/2,    Vs,    'V20 
Eesolve  into  two  binomial  factors  [see  P.  39]  : 

83.  a  —  b        85.  x^  —  y^        87.  16  —  a;        89.  a;^  —  5 

84.  x  —  4^        86.  ic3  —  1/3         88.  a;  —  25        90.  y  — 2 
Write  the  quotients  of  the  following  examples  [see  P.  45]: 

91.  {x-y)-r-  (x^  -  y^)  93.  (x-y)^  (x^  -  y^) 

92.  {x-{-y)-^  {x^  +  2/*)  94.  {x-\-y)^  {x^  +  ^^) 


MISCELLANEOUS  EXAMPLES. 


241 


96.  {l^x-S\y)^{2x^-^y^) 

97.  (16a;-8l2/)^(2a;i  +  32^*) 

98.  (x^  -  y^)  -^  {x^  -  y^) 

Reduce  to  lowest  terms  : 
99       ^  ~  *^ 

a;—  Vy 

1  no 

"""  ^x  +  ^y 

^^^-  V^+  Vy 

a—  Vb 

■  (Va:  -  Vy)  Va;  +  y 

x{x-\-y) 

101.    a/-   ,     8/- 

Va;+ Vy 

Va:^-y^ 
104.  1~= 

ic  Vic  +  y 

Expand  : 

105.  \x^-]-{xy)^-\-y^  \x^ 

■-(^y)*  +  y^l 

106.  (a  +  Va«  -  a:*)* 

109.  («  4-  5  V- 1)^ 

107.  (a  -  *  V-  1)' 

no.  (-2-3  V- 2)3 

108.  (VS+V^)* 

111.  (2  V2- V3)* 

Simplify  : 

112.  i/27  Vl35a«&* 

4  , T= 

115.  4/49  +  12  V5 

113.  V25a:*  Vy 

116.  V2a;-  V4a;2-4 

'"i/i/i 

117.  V-3-2  V2 

„0  a;4-A/^  ,  x-^- 

I_3/  _  yj^^^  p 

x—v  —  y      ar+v- 

-2^ 

119. 


Va;  +  y+Va:-y  _^  Va;  +  y- Va:-y  ^  ^^^^^  ^ 
'x-{-y  —  "s/x  —  y      Vx  +  y  +  Va;  —  y 


120.  Square 


/i+^ 


242  ELEMENTARY  ALGEBRA, 

121.  Square   a/ x a/xA-- 

Y  X        Y  ^ 

122.  Extract  the  square  root  of  : 

x^  +  2xy^-]-Sx^y^-{-2x^y-^y^ 

123.  Extract, the  cube  root  of  : 

a;f  +  3^  +  6^i+7  +  4  +  §  +  -3 


Radical 

Equations. 

Illustrations— 1.  Solve 

X^=4: 

(A) 

Solution:  Extract  ^J, 

xi=±2 

(1) 

Cube  (1), 

x=±8 

2.  Solve  V2a;2  =  2 

(A) 

Solution:  Cube  (A), 

2a;«  =  8 

(1) 

Divide, 

a;2  =  4 

(2) 

Extract  V, 

x=  ±2 

3.  Solve  Vx-\-14:-^Vx—14:=  14  (A) 
Solution : 

Square  (A),    a;  + 14  +  2  V^^-196  +  x-U  =  196  (1) 

Transpose,  2  ^x^  —  196  =  196  -  2  a;  (2) 

Divide,  ^x^  — 196  =  98  -  a;  (3) 

Square,  *  a;^  -  196  =  9604  -  196  a;  +  a;^     (4) 

Transpose,  196  a;  =  9800  (5) 

Divide,  a;  =  50 

Caution. — In  squaring  a  radical  binomial,  do  not  simply  square 
each  term.    Thus,  i\/x  x  'x/ af  is  not  x  +  a,  but  a;  +  2  \/ax  +  a. 

4.  Solve  Vx^ -{-ax  —  Vx  =  x  (A) 
Solution :  Transpose,     ^/x^  +  ax  =  x  +  ^/x  (1) 

Square,  x'^  -^  ax=.x^  ->r-2x  ^/x  +  x  (2) 

Transpose,  2  x  ^/x  =x{a  —  \)  (3) 

Divide  by  x,  2  v^  =  (a  —  1)  (4) 

Square,  4  a;  =  (a  —  1)*  (5) 

Divide,  X  =  - — T — - 


RADICAL  EQUATIONS, 


5.  Solye  a^ -  3 x -  6  V2^^-3x-S  =  -2  (A) 
Solution :  Subtract  3  from  both  members, 

(a;8_3a;_3)_6(a;2-3rc-3)i  =  -5  (1) 
Complete  the  square, 

(a;«-3a;~3)-6(        )  +  9  =  4  (2) 

Extract  V»              ^x^-dx-3-  3  =  ±  3  (3) 

Transpose,                      y\/x^  —  dx  —  3  =  5  or  1  (4) 

Square,                                2^  -  32;  -  3  =  25  or  1  (5) 

Transpose,                                 a;»  —  3  a;  =  28  or  4  (6) 

9      121        25 
Complete  the  square,        x^^Sx  +  ■j  =  -j-  or  -j-  (7) 

Extract  V,  a-  |=  ±  ^  or  ±  |     (8) 

Transpose,  a:  =  7,  —  4,  4,  or  —  1 

EXERCISE     128. 

Solve  : 

1.  V4^=16 

2.  (a; +  2)*  =  36 

3.  (3  x)i  =  9 

4.  V?=l 


7. 

^/x  — 

x-3 

Va;  + 

"4  =  5 

8. 

:  2  +  Va;  -  12 

9. 

10. 

flj  =  2  Va;  —  a 

11. 

=  7-V4cX-K 

6.  V(a;-a3)«  =  4a* 

6.  V(a;*  +  «T  =  «*  12.  V?+2"=  Va;-4 

13.  Va;4-2-f-  Va;-2  =  2 


14.  Vx-\-Vx  +  2=Vx-{-3 


15.  Va;  —  10  +  Va;  -  9  =  5 


16.  Va;  +  3-f  Vx-3  =  2  a/x-^^ 

,-   ^  +  1  ,   ^         /-  v^-f  2      ^ 

17.  — T— +5=Va;  19.     /-         =2 

V  a;  Va;  —  2 

18.  ^7=-  -  a  =  — W  20.  /-  =  i» 

Vx  Vx  a  —  Vx 


2U 


ELEMENTARY  ALGEBRA. 


21. 
22. 
25. 
26. 
27. 

29. 
30. 


l-\-^/l- 


1  _  Vl  -  a; 
ic  —  4 


=  5 


Va;-2 


=  4 


23. 


24. 


X  —  a 


\fx  —  Va 


2 

■  2Va 


Vx  +  l-{ 


Vx-i-1         3 


xJf-^x  +  1 


Vx-i-l 


Vax-\-a  r-  .      r- 

—j= '—^  =  ^x  -{-  Va 

x-\-a  v^  —  V« 


28.  -T  + 


-^i 


^  + 
o; 


31.  -7= 


Vx  —  Va 


32. 
33. 
34. 
35. 

36. 
41. 
42. 
43. 
44. 
45. 


Vx  +  Va  x-\-a 

4:X  —  9a  =  2Vx-\-SVa 

x-a=  Vx-\-  Va  37.  a;  +  4+ Va:  +  4  =  20 

Vl  —  a;  =  1  —  Vx  3Q.xi  —  x^  =  d 

xJ^x^  =  Q  39.  3xi  +  x^=z2 

« Of*  +  J a;"2' =  c  40.  a;^ +  0:4  =  2 


2x^-^Sx-4.V2a^-i-Sx-2  =  ll 
2ax  —  x^  =  2a^  —  aV2ax  —  x^ 


a^-4:X^-2Va^-4:a^-^4:  =  dl 
(x  -SY  =  13-{x^-ex-{- 16)^ 
x^  +  5  Va^-lQx=:  16  ic  +  300 


RADICAL  EQUATIONS.  245 

lsVx-2Vy=  0)  ^^'  \x-  Vy  =  Vy) 

Vx-\-y  +  Vx  —  y  =  S 
Vx-{-y  —  Vx  —  y  =  2 

Vx  4-  Vy  =  1 )  \x4-  Vx  +  V  =  11 

SO.    i      r^  ^  y  51.    ^  ^ 


49. 


4cx-{-  V 9 y  =  4:)  '  ly-{-  ^^ -i-y  =  4 


62. 


2^  -  ^^  +  ViT  -  2^2  _    20 
xy{xy-{-l)  =  240 
53.  x2  +  2a:  =  6  +  4  V3 


Character  of  the  Roots  of  Equations. 

Definitions. 

263.  A  root  containing  one  or  more  imaginary  terms  is 
an  imaginary  root  j  as,  x  =  a  ±b  V—  1. 

264.  A  root  containing  no  imaginary  terms  is  a  real 
root ;  as,  x  =  a-{-b,  or  x  =  S  ±  a/2. 

265.  A  real  root  that  contains  one  or  more  irrational 
terms  is  an  irrational  root  j  as,  x  =  3  ±  V2, 

266.  A  real  root  that  contains  no  irrational  term  is  a 
rational  root ;  as,  x  =  a-\-b,  or  a;  =  3  ±  a/J. 

SIGHT      EXERCI  SE. 

Tell  which  of  the  following  roots  are  real  and  which 
imaginary : 

1,  x=  vT+5  6.  x=  V53  -  7* 


2.  x=  V8-2  n.  x=  V5  (7  -  12) 


3.  x=  V5-9  Q.  x=  V-  3  (6  -  9) 


4.  x=  V12-3X5  9.  x=  V(- 2)2-1 

6.  x=  V42-8  10.  X  =  Vb"^  -  4« 


246  ELEMENTARY  ALGEBRA, 

If  a  and  h  are  positive,  and  a  is  greater  than  h 


11.  X  =  wa  —  h  14.  a;  =  V—  a{a  —  b) 

12.  x=  's/'b  —  a  15.  a;  =  a/^^  —  a  J 


13.  a:  =  -Ja(a  —  V)  16.  a;  =  wc^^—cFb 


Tell  which  of  the  following  roots  are  rational,  which 
irrational,  and  which  imaginary  : 

17.  x=  V18  +  7 


18.  x=  VSO  — 5 


19.  X  =  V15  —  3 


20.  a;  =  V32  -  2  X  5 


22. 

a;  = 
a;  = 

a:  = 
x  = 

:V8- 

-32 

23. 

:  ^/72- 

-13 

24. 

:  V4  X  5  - 

6X3 

25. 

:V5^- 

-32 

21.  x=Vs^  —  20  26.  a;  =  V25  -  4^ 


Give  the  sign  of  each  of  the  following  roots  : 

27.  a;  =  4+ V12  30,  x=-4:+^/5 

28.  a;  =  8  —  V96  31.  a;  =  8  —  Vlb 

29.  a;  =  -  2  -  ^  Vio  32.  a;  =  -  10  +  -  Vi20 

If  ^  and  g'  are  positive,  and  p^  is  greater  than  q  : 

33.  x=p-{-  ^p^-^q  36.  a;  =  —p  —  V^^  _|_  ^ 

34.  a;=^  —  VpH-^  37.  J9  4-  V^i?^  —  g! 

35.  a;  =  —p  +  v5M-~^  38.  — ^  -}-  Vp^  —  q 


The  roots  of  the  equation  a;^  +^  a;  =  g'  are 

in  which  j^^  is  the  square  of  -^  the  coefficient  of  x,  and 
q  is  the  absolute  term. 


CHARACTER  OF  THE  ROOTS  OF  EQUATIONS.    247 

Tell  the  character  of  the  above  roots,  whether  real  or 
imaginary : 

1.  If  p  and  q  are  positive. 

2.  If  p  is  negative  and  q  positive. 

3.  If  p  is  positive  and  q  negative,  and  q  <j^p^,  nu- 
merically. 

4.  If  J9  is  positive  and  q  negative,  and  q  >  -rp^y  nu- 
merically. 

5.  If  p  is  negative  and  q  negative,  and  q<-TP^,  nu- 
merically. 

6.  If  p  is  negative  and  q  negative,  and  q>  -rp^y  nu- 
merically. 

Tell  whether  the  above  roots  are  rational  or  irrational : 

7.  If  -7P^-\-q  is  a  perfect  square. 

8.  If  ^  =  0. 

9.  If  q  is  negative,  and  numerically  equal  to  -rp^, 

10.  If  jt?  =  0, 

Give  the  signs  of  the  above  roots,  and  tell  which  root  is 
numerically  the  greater  : 

11.  If  j9  and  q  are  both  positive. 

12.  If  p  is  negative  and  q  positive. 

13.  If  ^  is  positive,  q  negative,  and  -tP^>  q,  numeri- 
cally. 

14.  If  p  and  q  are  negative,  and  -tP^>  q,  numerically. 

16.  If  j9  and  q  are  negative,  and  -p^  =  qj  numerically. 

What  are  the  values  of  p  and  q  in  the  following  equa- 
tions : 

16.  a:2_|_4a.^7  xg,  a?-^xz=-b 

17.  ar-9a;  =  5  20.  ea:^  — 9  =  0 

18.  a^  +  6a;=-4  21.  2a:«  +  5a;  -  3  =  0 


248  ELEMENTARY  ALGEBRA. 

In  the  following  equations,  are  the  roots — 
1.  Real  or  imaginary  ?    2.  Rational  or  irrational  ?    3. 
Positive  or  negative  ?    4.  What  are  their  relative  values  ? 

22.  x^-\-Q>x=l  28.  a;2  +  6  a;  =  -  9 

2Z.  x^  —  4:X  =  b  29.  ic^  —  5 a:  =  10 

24.  x^-\-bx=  —  Q  zo.  x^-]-^x=—6 

25.a^  —  Sx=—2  3l.a^  —  6x=—8 

26.  x^-\-7x=  —16  32.  4:X^  —  l!x=  —  l 

27.^  +  1^  =  1  33,  0^-^=-^ 


Inequalities. 

I.   Definitions  and   Principles. 

267.  An  expression  denoting  that  two  quantities  are 
unequal  in  value  is  an  Inequality. 

268.  The  symbol  of  inequality  is  >,  read  greater  than; 
or  <,  read  less  than. 

269.  The  quantities  compared  in  an  inequality  are  the 
memhers  of  the  inequality. 

270.  Two  inequalities  are  said  to  subsist  in  the  same 
sense,  when  the  first  members  are  both  greater  or  both  less 
than  the  second  members. 

271.  Two  inequalities  are  said  to  subsist  in  an  opposite 
or  contrary  sense,  when  the  first  member  of  the  one  is  the 
greater  and  the  second  member  of  the  other. 

272.  A  negative  quantity  is  considered  less  than  a  posi- 
tive quantity,  whatever  their  absolute  values. 

273.  The  process  of  changing  the  form  of  an  inequality 
without  changing  its  sense  is  transformation. 


INEQUALITIES.  249 

274.  The  followiug  principles  of  transformation  may 
readily  be  illustrated  : 

Prin,  114, — 1.  The  same  or  equal  quantities  may  he 
added  to  both  members  of  an  i^iequality. 

2.  The  same  or  equal  quantities  may  be  subtracted  from 
both  members  of  an  inequality. 

3.  Both  members  of  an  inequality  may  be  multiplied  by 
the  same  or  equal  positive  quantities. 

J^  Both  members  of  an  inequality  may  be  divided  by 
the  same  or  equal  positive  quantities. 

6.  Two  unequal  positive  members  may  be  raised  to  the 
same  power. 

6.  Two  unequal  positive  members  may  have  the  same 
root  extracted,  provided  the  positive  results  only  are  comr 
pared. 

7.  The  sum  of  two  inequalities^  subsisting  in  the  sam^ 
sense,  may  be  taken  member  by  member. 

276.  {a  -bf>0  whether  a>b  ox  b>a[V.  27]. 
Expanding,  a^ -%ab-{-b^  >  0  (1) 

Add  'Zab  to  both  members  [P.  114,  1]  a^ -\- b"^  >  2 a b. 
Therefore, 

Prin,  115, — The  sum.  of  the  squares  of  two  unequal 
quantities  is  greater  than  twice  their  product. 

276.  a^-\-b^>2ab\V.  1161  (1) 

a2  +  c2>2ac        ''  (2) 

b^-]-c'>2bc        "  (3) 

Adding  member  by  member  [P.  114,  7], 

2a^-\-2b^-[-'-Zc^>2ab-\-2ac-\-2bc        (4) 
Dividing  by  2  [P.  114,  4], 

a'^-^b^-}-c'>ab-\-ac-\-bc. 
Therefore, 

Prin,  116, — The  sum  of  the  squares  of  three  unequal 
quantities  is  greater  than  the  su7n  of  their  products  taken 
two  and  two. 


250  ELEMENTARY  ALGEBRA, 


2.  Examples. 

Ulustration. — Which  is  the  greater,  a^ -{■  ¥  ov  a^  h -\- a  W, 

for  any  positive  values  of  a  and  h  ? 

Solution  :  a^  +  b^>  =  <aH  +  al^ 

Factoring,  (a  +  h){a^  —  ab  +  ¥)>  =  <ab{a  +  b) 

Dividing  hj  (a  +  b),  a^  —  a b  +  b^  >  =  <:,  ab 

Adding  a  &  to  both  members,        a^  +  &^  >  =  <  2  a  6. 

But  a'  +  b^>2ab  [P.  115], 

. • .     a^  +  b^^a^b  +  ab^,  since  no  operation  has  been  perf onned 

to  change  the  sense  of  the  inequality. 

EXERCISE     126. 

Prove  the  following  statements  true  for  unequal  positive 
values  of  the  letters  : 

1.  a^  +  ab^>2aH  4.  {a  +  hy>  4:an-i-4.ab^ 

2.  aH-{-a¥>an^  +  ab^        6.  a^  +  db^  >  2ab  +  2b^ 

3.  {a-\-by>4tab  6.  a^  >  a^ -i-a -1 

7.  a^  +  an^-ira^c^>aH  +  a^c  +  aHc        8.  a^>2a-l 
9.  U  a^-{-4:X>  12,  show  that  x>2 

10.  If  3a^-^6x>  42,  show  that  x>3 

11.  If  7x^  —  3x<  160,  show  that  x<6 

12.  -  +  ^>2  13.  -2  +  ^>-  +  - 
14.  7?y-\-xy^-\-Q?z-\-x^-\-y^%-\-yz^>^xyz 

a-\-b      a      ,  ^  a-\-b   ^  a      . 

15.  — h^  >  -,  when  a<c        16.  — -^  <  — ,  when  a>c 
c-\-b      c  c-\-b      c 

a  —  b  ^  cc      1 

17.  7  >  =  <  -,  when  a  <  =  >  c 

c  —  b  c 

18.  Which  is  the  2rreater,  ,  or    „      ,p,  if  a  and  h 
...      ^               ^          'a  —  b        a^  —  W 

are  positive  ? 

19.  What  integral  value  of  x  will  satisfy  3 ic^  +  ^a;  >  64 
and  3  ic2  +  4  ir  <  132  ? 


CHAPTER  VII. 

RATIO,   PBOPOBTIOJf,   AKB 
PROGRESSIOJ^. 


Ratio. 


I.    Definitions  and   Principles. 

277.  A  relation  of  values  exists  between  two  similar 
quantities — that  is,  one  of  them  is  a  number  of  times  or  a 
part  of  the  other. 

278.  The  relation  which  the  value  of  one  quantity  bears 
to  that  of  another  is  the  Ratio  of  the  quantities,  and  is 
obtained  by  dividing  the  quantity  compared  by  the  quan- 
tity with  which  it  is  compared. 

niustration. — The  ratio  of  3  apples  to  5  apples  is  — , 

3 

since  3  apples  =  ;^  of  5  apples. 

279.  A  ratio  is  expressed  by  writing  a  colon  between  the 
quantities  compared,  or  by  a  common  fraction. 

niustration. — The  ratio  oi  a  to  h  i^  a  :  h,  or  -r. 

280.  The  quantity  compared,  or  the  first  term  of  a 
ratio,  is  the  antecedent ;  and  the  quantity  with  which  the 
comparison  is  made,  or  the  second  term  of  the  ratio,  is 
the  consequent, 

281.  Since  the  ratio  is  obtained  by  dividing  the  quan- 
tity compared  by  the  quantity  with  which  it  is  compared, 
it  follows  that, 


252  ELEMENTARY  ALGEBRA. 

Prin.  117 > — The  ratio  equals  the  antecedent  divided  hy 
a 


trie  consequent ;  or  r 

c 

282.  Since  r  =  — ,  a=.c  X  r,  and  c  =  —.     Therefore, 

c  r  ' 

Prin,  118. — The  antecedent  equals  the  ratio  times  the 
consequent, 

Prin,  119,  —  The  consequent  equals  the  antecedent 
divided  hy  the  ratio, 

283.  Since  —  =  r,  —  =^nr,  and  =  nr  (Ax.  4). 

c  c  c-^  n  ^  ' 

Therefore, 

Prin,  120, — Multiplying  the  antecedent  or  dividing  the 
consequent  multiplies  the  ratio, 

284.  Since  -  =  r,    =  -,   and   —  =  -    (Ax.    5). 

c  en  nc      n    ^  ' 

Therefore, 

Prin,  121, — Dividing  the  antecedent  or  multiplying 
the  consequent  divides  the  ratio, 

285.  Since  -  =  r,  —  =  r,  and  — '- —  =  r.    Therefore, 

c  nc  c-^n 

Prin,  122, — Multiplying  or  dividing  loth  terms  of  a 
ratio  hy  the  same  quantity  does  not  alter  its  value, 

286.  The  product  of  two  or  more  simple  ratios  is  a 

compound  ratio.     Thus,    ]  ^  !  ^  [  ^^  f  X  ^  is  a  compound 
ratio. 

287.  To  duplicate  a  ratio  is  to  use  it  twice  in  a  com- 
pound  ratio.     Thus,  the  duplicate  of  a-.l  is  ?X^  =  t^. 

0  0         0^ 

288.  To  triplicate  a  ratio  is  to  use  it  three  times  in  a 
compound  ratio. 

Thus,  the  triplicate  of  a-.  I  is  -^X^X^  =  t^. 

0  0  0         0^ 


RATIO,  253 

289.  When  the  antecedent  and  consequent  of  a  ratio 
are  equal,  it  is  called  a  ratio  of  equality ;  as,  a  :  a,  or  1:1. 

290.  When  the  antecedent  is  greater  than  the  conse- 
quent, the  ratio  is  greater  than  one,  and  is  called  a  ratio  of 
greater  inequality. 

291.  AVhen  the  antecedent  is  less  than  the  consequent, 
the  ratio  is  less  than  one,  and  is  called  a  ratio  of  lesser  in- 
equality, 

292.  When  the  ratio  of  two  quantities  can  he  exactly 
expressed  hy  a  rational  number  or  fraction,  it  is  said  to  be 
commensurable, 

293.  When  the  ratio  of  two  quantities  can  not  be  exactly 
expressed  by  a  rational  number  or  fraction,  it  is  called  an 

incommensurable  ratio  ;  as,   a/2  :  a/3  —  i/ ~, 
2.  Examples. 

EXERCISE     127. 

3  5 

1.  Find  the  ratio  of  4  to  20  ;  16  to  12  ;  -^  to  9  ;  8  to  - ; 
2,3  56 

2.  Find  the  value  of  ab^  :  a^b  ;   a^  —  a^  :  a-\-x; 

{a  +  by-.a^-b^;  a^ -^7?  :  a""  -  ax-^3? 

3.  Find  the  value  of  : 

m-\-n    m^^n^a^  —  y^x  —  y  1       ^     a 

m  —  n'  (m  —  ny    x-^  y  '  a^-{-  y^^  7?  —  y^'  x  —  y 

4.  Reduce  to  their  lowest  terms  : 

25:75;  aH^  \  aV  \  a^^ab-.ab  +  b^ 
6.  Cleai*  of  fractions  and  reduce  to  lowest  terms  : 

j.3-,1     ^n3„^l      X     z  ,  a        ,   c 

2-r:7^;  18-i-:31-i-;   -:  —  ;   a -\- -r  :  c -{- j- 

4:       2  4         4:^   y    xy^  b  b 

12 


254  ELEMENTARY  ALGEBRA. 

6.  Compound  the  ratios  '4:5,  5  :  -6,  and  -03  :  40 

7.  Compound  the  ratios  2*5  :  -32,  -08  :  1*5,  and  -12  :  -016 

8.  Which  is  the  greater,  the  ratio  of  2—:  7—  or  the 
1     ..1  ^       ^ 


duplicate  ratio  of  2  —  :  7  -  ? 


9.  What  must  be  subtracted  from  both  terms  oi  a;h 
to  make  it  c:  d^ 

10.  Compound  the  ratios  of  : 

a-l''  {a^yf^^^    a-\-l)''  {a-  If 

11.  If  5  horses  and  8  cows  cost  as  much  as  8  horses  and 
2  cows,  what  is  the  relative  value  of  a  cow  to  a  horse  ? 

12.  Find  the  ratio  of  2  to  a/2  to  within  one  thousandth; 
also,  the  ratio  of  3  to  a/3. 

13.  The  side  of  a  square  is  4  feet.  What  is  the  approxi- 
mate ratio  of  the  side  to  the  diagonal  ? 

14.  If  the  same  number  be  added  to  both  terms  of  a 
ratio  of  lesser  inequality,  will  it  be  increased  or  diminished  ? 
Which,  if  the  same  number  be  subtracted  from  both  terms  ? 

15.  If  the  same  number  be  added  to  both  terms  of  a 
ratio  of  greater  inequality,  what  will  be  the  effect  ?  What, 
if  the  same  number  be  subtracted  from  both  terms  ? 

16.  The  ratio  of  A's  money  to  B's  is  the  same  as  the 
ratio  of  5  to  6,  and  they  together  have  $1320.  How  much 
has  each  ? 

17.  The  sum  of  A's  and  B's  ages  bears  the  same  relation 
to  A's  age  as  A's  age  bears  to  8  years,  and  the  difference 
of  their  ages  is  10  years.     Eequired  the  age  of  each. 

18.  The  fore-wheel  of  a  wagon  makes  128  revolutions 
more  in  going  a  mile  than  the  hind-wheel,  and  their  cir- 
cumferences are  in  the  ratio  of  5  :  6.  What  is  the  circum- 
ference of  each  ? 


PROPORTION,  256 

Proportion. 

Definitions. 

294.  The  equality  of  two  or  more  ratios  may  be  ex- 
pressed by  writing  between  them  a  double  colon,  or  the 
symbol  of  equality. 

Thus,  the  fact  that  the  ratio  of  2  to  3  equals  the  ratio 
of  4  to  6  may  be  expressed  : 
1.    2  :  3  :  :  4  :  6    \ 

2  _  4        >-  read  2  is  to  3  as  4  is  to  6. 
^*         3  "^  6        ) 

295.  The  expression  of  the  equality  of  two  or  more 
equal  ratios  is  called  a  Proportion. 

296.  A  proportion  of  two  simple  ratios  is  a  simple  pro- 
portion ;  one  of  three  or  more  ratios,  a  multiple  proportion. 

297.  The  ratios  of  a  proportion  are  called  couplets. 

298.  If,  in  a  multiple  proportion,  the  consequent  of 
each  couplet  is  the  same  as  the  antecedent  of  the  following 
couplet,  it  is  called  a  continued  proportion. 

Thus,  a'.hwh'.  c\\  c'.d  is  a  continued  proportion. 

299.  Every  simple  proportion  has  four  terms.  The  first 
and  fourth  are  called  the  extremes ;  the  second  and  third 
the  means ;  the  first  and  third  the  antecedents ;  and  the 
second  and  fourth  the  consequents. 

300.  A  mean  proportional  between  two  quantities  is  a 
quantity  to  which  the  first  bears  the  same  relation  that  the 
quantity  bears  to  the  second. 

Thus,  J  is  a  mean  proportional  between  a  and  c,  when 
a  :b  :  :  b  :  c. 

301.  A  third  proportional  to  two  quantities  is  a  quan- 


256  ELEMENTARY  ALGEBRA, 

tity  to  which  the  second  bears  the  same  relation  that  the 
first  bears  to  the  second. 

Thus,  c  is  a  third  proportional  to  a  and  I,  when 
a  :  h  :  :  1)  :  c. 

302.  If  a  proportion  contains  one  or  more  compound 

ratios,  it  is  a  compound  proportion. 

When  the  word  "  proportion  "  is  used  alone,  it  designates  a  simple 
proportion. 


Propositions. 


I.  In  any  proportion,  the  product  of  the  extremes  equals 
the  product  of  the  means. 

Given    a-.h  ::  c  :  d  :  (A) 

Prove    aX  d=bx  c 

Demonstration  :         T  =  T  [another  form  for  (A)], 
Clear  of  fractions,     a  x  d  =  b  x  c. 

CorcUary  1, — Either  extreme  equals  the  product  of  the 
means  divided  ly  the  other  extreme. 

dyr,  2, — Either  mean  equals  the  product  of  the  extremes 
divided  hy  the  other  mean. 


II.  If  the  product  of  two  quantities  equals  the  product 
of  two  other  quantities,  either  pair  may  ie  made  the  ex- 
tremes, and  the  other  pair  the  means,  of  a  proportion. 


Given    m  X  n=p  X  q 

(A) 

Prove,    1.  m: p  '.'.  q:  n 

6.  p  :  m: 

:  n:  q 

2.  m:  q\:  p  \  n 

6.  p  :  n: 

:  m:  q 

3.  n:p  ::  q:m 

7.  q:m: 

:  n  :  p 

4.  n:  q\:  p'.m 

8.  q  :  n: 

:  m  :p 

Demonstration:  1.  Divide  (A)  by 

P  X  g 

(1) 

Divide  (1)  by  jp, 

m      q 
p-n 

(2) 

PROPOSITIONS.  257 

Write  in  another  form,  m:  p  ::  q:  n. 
Let  the  pupil  derive  the  remaining  seven. 

Exercise. — Write  the  eight  proportions  deducible  from  : 
3X4  =  2X6;   aXd=zhXc\  xXz=vXy 


303.  A  proportion  is  taken   by  alternation  when  the 
means  or  the  extremes  are  made  to  change  places. 

III.  If  four  quantities  are  in  proportion,  they  are  also 
in  proportion  by  alternation. 
Given    a:h  ::  c  -.  d  (A) 

Prove,    1.  a:  c'.'.h  :  d  2.  d  :  b  : :  c  :  a 

Demonstration :  a  x  d  =  c  x  b  [P.  I], 

Exercise. — Write  by  alternation  : 

3:4::9:12;  x  :  y  ::  m:  n;  x  :  a::  y  :b 


304.  A  proportion  is  taken  by  inversion  when  the  means 
are  made  the  extremes  and  the  extremes  the  means. 

IV.  If  four  quantities  are  in  proportion,  they  are  also 
in  proportion  by  inversion. 
Given    a:  b  ::  c  :  d 

Prove,    1.  b  :  a  : :  d  :  c  3.  c:  a::  d:b 

2.  b  :  d  : :  c  :  a  4:.  c  :  d::  a:b 

Demonstration :  a  x  d  =  b  x  c, 

ll'-V'-'^'-'l   and   \''-y-^'-l\   [P.iq. 

(b  :d::  a:c  S  ic:d::a:b  )    *-        -* 

Exercise. — Write  by  inversion  :    5  :  10  : :  15  :  30  ; 
X  :  m  : :  n  :  y  ;  a-^-b  :  a  —  b  : :  c-\-  d  :  c  —  d 


305.  A  proportion  is  taken  by  composition  when  the 
sum  of  the  two  terms  of  each  couplet  is  compared  with 
either  the  antecedent  or  the  consequent  of  that  couplet. 


258  ELEMENTARY  ALGEBRA. 

V.  If  four  quantities  are  in  proportion,  they  are  also 
in  proportion  by  composition. 

Given    a:  b  : :  c  :  d  (A) 

Prove,    1.  a-{-b  :  b  :  :  c-\-d  :  d 
2.  a-{-b  :  a  : :  c-\-d  :  c 

Demonstration :  -r  =  -^  [another  form  of  (A)] 

Add  1  to  both  members,  -r-  +  1  =  -r  +  1 

Reduce  to  improper  fractions,  — j—  =      ,     ;  or 

a  +  b  :b  :  :  c  +  d  :  d 
Let  the  pupil  prove  the  second  part. 

Exercise. — Write  by  composition  : 

2:3::6:9;   8:2::16:4;   x:a::y:b 


306.  A  proportion  is  taken  by  division  when  the  differ- 
ence of  the  two  terms  of  each  couplet  is  compared  with 
either  the  antecedent  or  the  consequent  of  that  couplet. 

VI.  If  four  quantities  are  in  proportion,  they  are  also 
in  proportion  by  division. 

Given    a-.bwc'.d 
Prove,    1.  a  —  b\b::c  —  d'.d 
2.  a  —  b'.awc  —  d'.c 

Demonstration  :  'b~'d  [^^^ther  form  of  (A)] 

Subtract  1  from  both  members,  ^  —  1  =  -y  —  1 

'  0  d 

-r,!  a  —  he  —  d 

Reduce,  — v —  =  —i —  ;  or 

a  —  h'.h'.-.c  —  d-.d 

Let  the  pupil  prove  the  second  part. 

Exercise. — Write  by  division  : 

3:9::6:18;   6:3::  12:6;  a-\-X'.  xwb^y  :  y 


PROPOSITIONS,  259 

307.  A  proportion  is  taken  by  composition  and  division 
when  the  sum  of  the  two  terms  of  each  couplet  is  compared 
with  the  difference  of  these  terms. 

VII.  If  four  quantities  are  in  proportio7i,  they  are  also 
in  proportion  hy  composition  and  division. 
Given    a:b  w  c  \  d  (A) 

Prove    a-\-b:a  —  h'.'.c-\-d'.c  —  d 
Demonstration  :  Take  (A)  by  composition,  — r—  =  —-r-  (1) 

Take  (A)  by  division,  ^^^  =  ^^         (2) 

Divide  (1)  by  (2),  ^  =  ^^d  5  ^^ 

a  +  b  :  a  —  b  :  :c  +  d  :c  —  d 
Exercise. — Write  by  composition  and  division  : 
3:8::  12:  33;   x  :  y  :  :  m, :  n  ;  x  — y  :  x-^  y  :  :  3  :  6 


VIII.  If  two  proportions  have  a  couplet  in  each  the 
same,  the  remaining  couplets  form  a  proportion. 
b::c:d  (A) ^ 

:c:d  (B)  ) 

:e:f 


Given    ,        ^ 


r,; 


Prove      a  :  b 


Demonstration:  -r- = 


y  =  ^(A)and^  =  j(B) 

-r  —  -f  (Ax.  1);  whence 
o  :  6  : :  e  :/ 

Exercise. — Prove  that, 

Cwr,  1, — If  two  proportions  have  the  antecedents  alike, 
the  consequents  form  a  proportion  ;  or, 

Given    a-.bw  c\  d  and  a\x\\  c.y 

Prove    b  '.  d  ::  x:  y 

Cor,  2. — //'  two  proportions  have  the  consequents  alike, 
the  antecedents  form  a  proportion  ;  or. 

Given    a:b\:c:d  and  x:b\\y  \d 

Prove    a:  c'.'.x:  y 


260  ELEMENTARY  ALGEBRA, 

Car.  3, — If  two  proportions  have  a  QOuplet  in  propor- 
tion, the  remaining  couplets  form  a  proportion  ;  or. 
Given     a  :  h  :  \  c  :  d,  e  -.  f :  :  g  :  h,  and  a:h::e:f 
Prove    c\  dwg  \h 

308.  Equimultiples  of  two  or  more  quantities  are  the 
products  obtained  by  multiplying  each  of  the  quantities 
by  the  same  number. 

IX.  Equimultiples  of  two  quantities  are  proportional 
to  the  quantities  themselves. 

Given  the  two  quantities  a  and  5  and  their  equimul- 
tiples 7na  and  mh, 

Prove    ma'.mh'.-.a'.h 

_  ...        ma      a 

Demonstration  :  — r  =  -r 

mb       0 

ma  :  mb  :  :  a  :  b 

Exercise. — JProve  that  equal  parts  of  two  quantities  are 

proportio7ial  to  the  quantities  themselves  ;  or  that, 

a     h  -, 

—  :  —  ::  a:o 
m    m 


X.  If  four  quantities  are  in  proportion,  equimultiples 
of  the  first  couplet  are  proportional  to  equimultiples  of  the 
second  couplet. 

Given    a\l\\c\d  (A) 

Prove    ma:  mh  ::  nc:  nd 

Demonstration  :      x  =  t  [another  form  of  (A)] 

a  ma  ^  c  nc 
T-  =  — r>  and  -=-  =  —-5 
b      mb  a      nd 

ma_nc 

mb~  nd^ 

ma :  mb  :  : nc : nd 

Exercise. — Prove  that, 

1.  Equal  parts  of  the  first  couplet  are  proportional  to 
equal  parts  of  the  second  couplet. 


PROPOSITIONS.  261 

2,  Equimultiples  of  the  antecedents  are  proportional  to 
equimultiples  of  the  consequents, 

S.  Either  extreme  may  he  multiplied  and  the  other 
divided  by  the  same  quayitity, 

4,  Either  mean  may  he  multiplied  and  the  other  divided 
by  the  same  quantity, 

XI.  If  two  quantities  are  increased  or  diminished  by 
like  parts  of  themselves,  the  results  are  proportional  to  the 
quantities  themselves. 

Given  the  two  quantities  a  and  h,  to  prove, 

1.  a-\ —  a  :  h-\ —  h  : :  a  :  h 
n  n 


m       ^      m  , 

2.  a a:  h h  ::  a:b 

n              n 

Demonstration:  a\\  ±  -) 

\         n)       a 

a  ±.  —  a 

n          a 

^{^-t)~' 

n 

a±^o:ft± 

™6, 

n 

::a:b 

Exercise. — Given    a:  h  ::  c  :  d 


Prove    a±—a:b±—h::c±-c:d±-d 
n  n  a  a 


XII.  If  four  quantities  are  in  proportion,  like  powers 
and  like  roots  of  them  are  also  in  proportion. 
Given    a\h\\c\  d  (A) 

Prove,    1.  «" :  J" :  :  c"  :  d^ 

2.    Va  :  Vh  :  :  ^/c  :  Vd  .      ^ 

Demonstration :  a  x  d  =  b  x  c  [F.  I]  (1) 

liaise  lx)th  members  to  the  nth  power, 

arxd*  =  h*xr*  (2) 

Then,  oT  :  h"" : :  c*  :  dT  [P.  II]  (3) 

Extract  the  nth  root  of  (1),    V »  x  V^  =  V^  x  aA  (4) 

Then,  V»  •  V^ : :  Vc :  Vd  (5) 


2G2  ELEMENTARY  ALGEBRA. 

XIII.   The  corresponding  memhers  of  two  equations  form 
a  proportion. 

Given    a —  I  (A)  and  c  =  d  (B) 
Prove     a\  c  :\i  :  d 

Demonstration  :  Divide  (A)  by  (B),    —  =  -^  ;  (1) 

or  a  :  c  : :  b  :  d 


XIV.   The  products  or  quotients  of  the  corresponding 
terms  of  tivo  jjroportions  form  a  proportion. 

\a:h'.:c\d  ( A)  ) 

Given    '  ^      ' 


e:f'.:g:h  (B)  S 

Prove,    1.  a  X  e:i  Xf:  :  c  X  g  :  dXh 
^    a     h       c     d 

^  f     g   h 

Demonstration :  a  x  d  =  b  x  c  (1),  and  e  x  h—.fx  g  (2)  [P.  I] 
Multiply  (1)  by  (3),  {a  x  e)  x  (d  x  h)z={b  x  f)  x  {c  x  g)  (3) 
Therefore,  a  x  e  :  b  x  f :  :  c  x  g  :  d  x  h  [P.  IIj 

Let  the  pupil  prove  the  second  part. 


XV.  If  two  proportions  have  three  terms  of  the  one 
equal  to  three  terms  of  the  other y  each  to  each,  the  fourth 
terms  are  also  equal. 

(B)i 


and  X  =  --^  [P.  I,  Cor.  1] 


Given   < 

,  a  : 

:  c  : 
;  c  : 

d 

X 

Prove      X  - 

=  d 

Demonstration 

l:  d: 

_b 

X  c 
CL 

Therefore, 

X  : 

=  d 

XVI.  In  any  multiple  proportion  the  sum  of  the  ante- 
cedents is  to  the  sum  of  the  consequents  as  any  antecedent 
is  to  its  consequent. 

Given     a  :h  -.:  c  :  d  :\  e  :f 

Prove     a-\-c-{-  c  :h-[-d  +/ '.'.  a-.h 


PROPOSITIONS.  263 

Demonstration  :  Let  r  equal  the  ratio  of  each  couplet, 

Then,  t-  =  ^>  Z"**'  ^^^  7~^ 

Clear  of  fractions,  a  =  br  (1),  c  =  dr  (2),  and  e=fr  (3) 

Add  (1),  (2),  and  (3),  a  +  c  +  e  =  {b  +  d  +  f)r  (4) 

Divideby6  +  rf+/,  ^±±±j=r  =  ^  (5) 

Therefore,  a  +  c  +  e  :  b  +  d  +  f : :  a  :  b 


XVII.  A    mean  proportional   between   two   quantities 
equals  the  square  root  of  their  product. 

Given  h,  a  mean  proportional  between  a  and  c,  to  prove 
h  =  Va  c. 

Demonstration  :  a:b  ::b  -.c  [317] 

.  • .    J'*  =  a  6  [P.  I]  ;  whence  b  =  ^^/o^ 


Additional  Propositions. 

EXERCISE     128. 

If  a-.h'.'.C'.dy  prove  that : 

1.  2fl:3J::2c:36?  Z.  na\  ml)  w  nc.md 

2.  3a:4^::6c:8c?  ^  a-\-  c\  a\\h-{-d'.  c 

5.  2a  +  3*:2c  +  3c?::2a:3J 

6.  w  a  -|-  wi  5  :  w  c  -f-  ''I  fi?  * :  «  :  c  7.  a:  d::hc:  d^ 

9.  (a  +  c)  2r :  (a  —  c)  a: :  :  (J  +  <Z)  y  :  (5  —  e/)  ?/ 
If  a  :  h  :  \  b  :  c,  prove  that  ; 

10.  a  :  c  :  :  a^  :  J*  ll.  Z*-  :  6-  : :  a  :  c 

12.  a:a  +  i::a  —  i:a  —  c 

13.  a  —  c:b  —  c  :  :b  -{-c  :  c 
Clear  of  fractions  : 


264-  ELEMENTARY  ALGEBRA. 

3    Solution  of  Equations  and   Proportions. 
Illustrations. — 

1.  Given  -^^,  =  77  to  find  the  value  of  x,         (A) 

Solution :  Write  the  equation  in  the  form  of  a  proportion, 

a;  +  4:a:-4::8:7  (1) 

Take  (1)  by  composition  and  division, 

2a- :8::  15:1  (2) 

Divide  first  couplet  by  2,    a; :  4  :  :  15  :  1  (3) 

a;  =  4x  15  =  60. 


/     .   7.^2  \J\C  to  find  X  and  y, 

{a  +  bf  (B)  ^  ^ 


2.   Given 

a/^  +  ^y  '  v^  —  Vy  :  :  a  :  b      (A) 

xy 

Solution :  Take  (A)  by  composition  and  division, 

2  V^  '  2  Vy  ::«  +  *:«  —  &  (1) 

Divide  the  first  couplet  by  2, 

^/x  :  Vy  ::«  +  &:  a  -  &  [P.  122]        (2) 
Square  the  terms,        x  :  y:  :  (a  +  bf  :  {a- b)^  [P.  II]  (3) 

Multiply  the  first  couplet  by  y, 

xy:y^::{a  +  bf  :  (a  -  bf  [P.  122]        (4) 
Substitute  (B)  in  (4), 

(a  +  bf:7f::(a  +  bf  :  {a  -  bf  ^  (5) 

Divide  the  antecedents  by  (a  +  bf, 

l:y^::l:{a-bf  [P.  122]       (6) 
y^z=(a-bf 
y  =±(a-&) 
Substitute  the  value  of  y  in  (3), 

X  \  ±{a-b)  '.'.{a  +  bf  :  {a-  bf 
Divide  the  consequents  by  (a  —  6), 

x\±\\\{a^bf\{a-b) 

a  —  b 

EXERCISE     129. 

Solve : 
1.  a::ar  +  6::4:7  2.  a:+6:a;-6::4:l 

3.  Vi  +  VS  :  V^  : :  VS  +  5  :  5 

4.  \/x  -\-  \fa  :  Vx  —  Va  :  :  Va  -f-  Vb  :  Va  —  Vb 


SOLUTION  OF  EQUATIONS  AND  PROPORTIONS.    2G5 

b.7^-a^  :  a^  -b^::x^-{-  a?  :  a^  +  J« 

6.  7?  —  a^  \  X  ->c  a  w  X  -\-  a  -rl 

7.  Va:*  —  c?  :  ^/x  —  a  : :  Vx -\-a  :  x  Vx 


Vx  +  V2       V2  4-1 


11.    {x-i-y:x::5:4:) 


Vx-j-S-{-Vx  —  S       1       12.    {x-\-y:x~y::3:l) 


Vx-[-a+Vx-a  _  ^       13.  a^-^f  -,  x^-f  : :  35  :  19 
Vx-i-a-Vx-a  ar-f/  =  52 


4    Examples  involving  Proportion. 

Ulnstration, — 1.  A's  age  is  to  B's  as  2  to  3  ;  but  in  10 
years  their  ages  will  be  to  each  other  as  3  to  4.  Required 
the  age  of  each. 

Solution :   Let  2  rr  =  A's  age ;  then  will 

Sx  =  B's  age ;  and 
2a;  +  10  =  A's  age  10  years  hence;  and 
dx  +  10  =  B's  age  10  years  hence ; 
then  23;  +  10  :  3  j:  +  10  : :  3  :  4  (A) 

8a;  +  40  =  9a;  +  30  (1) 

X  =  10 
2 2:  =  20,  A's  age; 
3  a;  =  30,  B's  age. 

2.  The  sum  of  A's  and  B's  capital  is  to  the  difference 
of  their  capitals  as  9  to  5  ;  but  if  A  withdraws  $100  and 
B  adds  1100,  their  capitals  will  be  to  each  other  as  4  to  3. 
Required  the  capital  of  each. 

Solution :  Let  x  =  A's  capital 

and  y  =  B's ; 

then         X  ■{-  y  :  x  —  y  :  :9  :  6,  (A) 

and   a;-  100  :  y  +  100  : :  4  :  3  (B) 

Solre  (A)  and  (B),      x  =  $423  ^3 ,  y  =  $107  ^ 


266  ELEMENTARY  ALGEBRA. 

EXERCISE     ISO. 

1.  The  length  of  a  room  is  to  its  width  as  4  to  3,  and 
the  floor  contains  588  square  feet  of  boards.  What  are 
the  dimensions  of  the  room  ? 

2.  A  man's  age  is  to  his  wife's  as  6  to  5  ;  but  30  years 
ago  his  age  was  to  hers  as  7  to  5.  Kequired  the  age  of 
each. 

3.  The  difference  of  two  numbers  is  to  the  difference  of 
the  squares  of  the  numbers  as  1  to  13,  and  the  product  of 
the  numbers  is  42.     Find  the  numbers. 

4.  A  and  B  are  in  partnership.  A's  capital  is  to  the 
whole  capital  as  5  to  8  ;  but  if  A  withdraws  $2000  and  B 
adds  $2000,  A's  capital  will  be  to  the  whole  capital  as 
3  to  5.     Required  each  man's  share  of  the  stock. 

5.  The  length  of  a  rectangular  field  is  to  its  width  as 
5  to  4 ;  but  if  4  rods  be  added  to  the  length  and  5  rods 
to  the  width,  they  will  be  to  each  other  as  6  to  5.  Find 
the  area. 

6.  Two  thirds  of  A's  money  is  to  ^4  of  B's  as  5  to  6, 
and  2/3  of  A's  +  %  of  B's  is  $1500.  Required  the  fortune 
of  each. 

7.  The  rate  of  a  fast  train  is  to  that  of  a  slow  train  as 
5  to  3,  and  if  it  is  60  miles  behind  the  slow  train  it  will 
overtake  it  in  3  hours.     What  is  the  rate  of  each  train  ? 

8.  I  have  a  cubical  box,  such  that  if  each  of  its  dimen- 
sions be  increased  by  one  foot  the  contents  will  be  to  the 
entire  surface  as  1  to  2.     Required  the  contents. 

9.  The  circumferences  of  circles  are  to  each  other  as 
their  diameters.  If  the  circumference  of  a  circle  whose 
diameter  is  one  is  7r  =  3*1416,  what  is  the  circumference 
of  a  circle  whose  diameter  is  ^  ? 

10.  The  areas  of  circles  are  to  each  other  as  the  squares 
of  their  diameters.     If  the  area  of  a  circle  whose  radius  is 


EXAMPLES  INVOLVING  PROPORTION.  267 

one  is  tt,  what  is  the  area  of  a  circle  whose  radius  is  r  ? 
What,  when  r  =  4  ? 

11.  The  surfaces  of  spheres  are  to  each  other  as  the 
squares  of  their  diameters.  If  the  surface  of  a  sphere 
whose  radius  is  one  is  4  tt,  what  is  the  surface  of  a  sphere 
whose  radius  is  r  ?    What,  whea  r  =  5  ? 

12.  The  volumes  of  spheres  are  to  each  other  as  the 
cubes  of  their  radii.  If  the  volume  of  a  sphere  whose 
radius  is  one  is  Y3  tt,  what  is  the  volume  of  a  sphere  whose 
radius  is  r  ?    What,  when  r  =  Q^ 

Surfaces  and  volumes  that  have  the  same  shape  are  similar.  To 
have  the  same  shape,  they  must  have  their  corresponding  angles  equal 
and  their  corresponding  dimensions  proportional. 

13.  Similar  surfaces  are  to  each  other  as  the  squares  of 
their  like  dimensions.  If  a  field  a  rods  long  contains  m 
acres,  what  will  a  similar  field  c  rods  long  contain  ? 

14.  Similar  volumes  are  to  each  other  as  the  cubes  of 
their  like  dimensions.  If  a  keg  whose  bung  diameter  is  c 
inches  holds  n  gallons,  what  will  a  similar  keg  d  inches  in 
bung  diameter  hold  ? 

15.  The  quantities  of  water  that  flow  through  circular 
pipes  are  to  each  other  as  the  squares  of  the  diameters  of 
the  pipes.  If  c  gallons  flow  through  a  pipe  m  inches  in 
diameter  in  one  minute,  how  many  gallons  will  flow 
through  a  pipe  n  inches  in  diameter  in  the  same  time  ? 


Limiting  Ratios. 

Definitions  and  Principles. 

309.  A  quantity  that  retains  the  same  value  throughout 
an  operation  or  discussion  is  a  constant. 

310.  A  quantity  that  continuously  changes  its  value — 


268  ELEMENTARY  ALGEBRA. 

that  is,  passes  from  one  value  to  another  by  successively 
assuming  all  values  lying  between  them— is  a  variable. 

lUnstration. — A  line  a  foot  long  is  a  constant.  A  line 
traced  by  a  point  moving  according  to  some  well-defined 
law  is  a  variable. 

311.  A  finite  unit  is  a  unit  of  comprehensible  size  or 
value. 

312.  A  quantity  that  can  be  expressed  in  finite  units  is 
afi?iite  quantity. 

313.  A  quantity  too  small  to  be  expressed  in  finite  units 
is  said  to  be  infinitely  small.  An  infinitely  small  variable 
is  called  an  infinitesimal,  and  may  be  expressed  by  the 
character  o ,  read  an  infinitesimal  or  zeroid. 

314.  A  quantity  too  large  to  be  expressed  in  finite  units 
is  said  to  be  infinitely  large.  An  infinitely  large  variable 
is  called  an  infinite,  and  may  be  expressed  by  the  char- 
acter a  ,  read  an  infinite. 

316.  The  entire  absence  of  quantity  is  called  zero,  and 
is  expressed  by  the  character  0,  read  zero. 

316.  The  unlimited  whole  of  quantity,  or  rather  un- 
limited quantity,  is  called  infinity,  and  is  expressed  by  the 
character  oo,  read  infinity. 

317.  If,  in  the  fraction  — ,  x  decreases  by  a  constant 

ratio  until  it  becomes  an  infinitesimal  and  a  remains  a 
finite  constant,  the  value  of  the  fraction  decreases  in  the 
same  ratio  [P.  55],  and  becomes  an  infinitesimal. 

Therefore, 

Brin.  123,  —  =  o .  An  infinitesimal  divided  ly  a 
finite  constant  is  an  infinitesimal. 


LIMITING  RATIOS,  269 

318.  Since  —  =  o ,  it  follows  that, 

a 

JPHn.  124,    o  X  a  =  o .    An  infinitesimal  multiplied 
by  a  finite  constant  is  an  infinitesimal, 

319.  Since  o  x  «  =  o ,  it  follows  that, 

PHn,  125,     —  =  a.     An  infinitesimal  divided  by  an 
infinitesimal  may  be  any  finite  constant, 

320.  If,  in  the  fraction  -,  x  increases  by  a  constant 

ratio  until  it  becomes  an  infinite  and  a  remains  a  finite 
constant,  the  value  of  the  fraction  increases  in  the  same 
ratio  [P.  54],  and  becomes  an  infinite.     Therefore, 

JPrin,  126,     —  =  a  .     An  infinite  divided  by  a  finite 

constant  is  an  infinite. 

321.  Since  —  =  a ,  it  follows  that, 

a 

Prin,  127,     oc  X  a=  cc.     An  infinite  multiplied  by  a 
finite  constant  is  an  infinite. 


322.  Since  a  X  a  =  oc ,  it  follows  that, 

Prin,  128,    —  -=.  a.    An  infinite  divided  by  an  infinite 
mxiy  be  any  finite  constant. 

323.  If,  in  the  fraction  -,  x  decreases  by  a  constant 

X 

ratio  until  it  becomes  an  infinitesimal  and  a  remains  a 
finite  constant,  the  value  of  the  fraction  increases  in  the 
same  ratio  [P.  54],  and  becomes  an  infinite.     Therefore, 

Prin,  129,    —  =  a  .     A  finite  constant  divided  by  an 
infinitesimal  is  an  infinite. 


324.  Since  —  =  a ,  it  follows  that, 

o 


Prin,  130,     o  X  oc  =  a.     The  product  of  an  infini- 
tesimal and  an  infinite  may  be  any  finite  constant. 


270  ELEMENTARY  ALGEBRA. 

325.  Since  o  x  oc  =  ^5,  it  follows  that, 

Prin,  131,  —  =  o .  A  finite  constant  divided  hy  an 
infinite  is  an  infinitesimal, 

326.  Since  — ,  — ,  and  a  X  o  are  each  satisfied  by 
any  finite  constant,  they  are  symbols  of  indetermination, 

327.  The  limit  of  a  variable  is  a  value  which  the  vari- 
able continually  approaches  but  which  it  can  never  reach, 
but  may  be  made  to  differ  from  it  by  less  than  any  assign- 
able quantity. 

Illustration. — If  a  point  starts  at  A  in  the  direction 
of  B,  and  goes  Yg  the  distance  the  first  second,  %  the 
remaining  distance  the  next,  Yg 

the  remaining  distance  the  third,       A \ j — j — B 

and  so  on,  the  distance  passed 

over  constantly  approaches  the  distance  from  A  to  B,  and 
will  eventually  differ  from  this  distance  by  an  infinitesi- 
mal, but  it  can  never  equal  this  distance.  From  A  to  B 
is  therefore  the  limit  of  the  distance  the  point  can  go. 

328.  The  limit  of  a  variable  that  decreases  by  a  con- 
stant ratio  is  zero. 

niustration. — If  Ys  a  line  be  cut  off,  then  Ys  the  re- 
mainder, and  so  on  indefinitely,  the  part  retained  continu- 
ally approaches  zero,  from  which  it  will  eventually  differ 
by  less  than  any  assignable  quantity.  Therefore,  zero  is 
the  limit  of  the  remainder. 

329.  A  variable  quantity  that  increases  by  a  constant 
ratio  has  no  limit.  This  fact  is  sometimes  expressed  by 
saying  that  its  limit  is  infinity. 

330.  A  function  of  a  variable  quantity  is  any  expression 
that  contains  the  variable. 

Thus,  ax^  -{-b  is  a  function  of  x. 


LIMITII^O  RATIOS.  271 

331.  A  function  of  a  variable  is  generally  a  variable  also. 
It  is  then  called  the  dependent  variable,  and  the  variable 
upon  which  it  depends  the  independent  variable. 

332.  The  limit  of  a  function,  when  the  independent 
variable  approaches  its  limit,  may  be  zero,  infinity,  or  a 
finite  quantity. 

Illustration. — 1.  If  x  approaches  a  as  a  limit,  the  func- 
tion  approaches  — ,  or  0  as  a  limit. 

2.  If  X  approaches  a  as  a  limit,  the  function 


X  —  a 
a 


approaches  — ,  or  oo   as  a  limit. 


X 

3.  If  X  approaches  a  as  a  limit,  the  function  — , — 
^^  x-j-a 

approaches  ,  or  -  as  a  limit. 

333.  Sometimes  a  factor  whose  limit  is  zero  is  common 
to  both  terms  of  a  fraction.     The  limit  of  the  fraction  will 

then  assume  the  irreducible  form  —  •      The  true  limit  is 

then  found  by  removing  the  common  factor  before  passing 
to  the  limit. 

Illustration. — If  x  approaches  a  as  a  limit,  the  func- 

2r*  ^3  Q 

tion  "2 2  approaches  —  as  a  limit.     This  form  results, 

because  the  common  factor  x  —  a  has  0  for  its  limit.     Re- 
moving  this  factor,  we  have  — -   -^ ,  which  has  for 

its  limit    ' ; ' =  X —  =  -  «. 

a-\-a  2a       'Z 

x^  _  ^3        3 

334.  To  express  that  the  limit  of  ^-^i  is  „  ^  ^^^^ 

the  limit  of  a;  =  a,  we  write  : 
Lim.  ^--2  (a:  =  «)  =  -«. 


272  ELEMENTARY  ALGEBRA, 


Examples. 

/v3  __  ^^3 

ninstrations. — 1.  Find  Lim. ' (x  =  a), 

^   (X^  Q^    q3 

Solution  :  =:x'^  +  ax  +  a^,        . • .    Lim.  (x  =  a)  = 

x—a  x—a^' 

Lim.  x^  +  ax  +  a\  {xz=a)  =  a^  +  a?  +  a^  =  3 ^^2, 
%  Find  Lim.      ~    (:r  =  00 ). 

X 

«...        x  —  a.a                                         x  —  a. 
Solution :  =  1 ,  .  • .    Lim.  te  =  00 )  = 

X  X  X       ^  ' 

Lim.  1  - -,  (ic  =  00)  =  1  -  ^- =  1  -  0  =  1. 

X    ^  '  00 


EXERCISE     181. 

Find: 

/y;*  //<*  ly 

1.  Lim. (x  =  «)  9.  Lim.  — -—  (a:  =  00  ) 

X  —  a  ^          '  x-\- 1  ^            ^ 

2.  Lim.    ^~     (a:  =  1)  10.  Lim.        f^^ ix  =  0) 

a;     ^          ^  ?72  a;^  +  w  a;  ^          ' 

3.  Lim.     p  , —  (a;  =  1)  11.  Lim. (x  =  a) 

XT  -\-  x  ^          '  X  —  a 

4.  Lim.  7 r^  (x  =  a)  12.  Lim. (x  =  1) 

(x  —  af^          '  x  —  \^          ' 

5.  Lim. {x  =  1)  13.  Lim.  -7—  (^  =  —  «) 

6.  Lim.  0^2  _^      (^  ^  0)  14.  Lim,  -^  (a-  =  0) 

x  —  a  axA-bx  ,            . 

T-.       rr^  —  a^ ,          ,  T .       ax-\-x4-l,            . 

8.  Lim.  — ; —  (a;  =  a)  16.  Lim.  ' —  (^  =  go  ) 

x-^a  ^          '^  X         ^           ' 


X  —  V'^  —  a^ , 
17.  Lim.  /  „  (x  =  a) 

,0    T-       x^  +  a^:^  +  a\  . 


ARITHMETICAL  PROGRESSIONS,  273 

Arithmetical  Progressions. 

Definitions  and   Principles. 

335.  Any  number  of  quantities  that  increase  or  decrease 
according  to  a  law  constitute  a  Series,  or  Progression. 

336.  The  quantities  which  compose  a  series,  or  progres- 
sion, are  call^  the  terms, 

337.  A  progression  in  which  each  term  after  the  first  is 
derived  from  the  preceding  term  by  the  addition  of  a  con- 
stant quantity,  is  an  arithmetical  progression, 

338.  The  constant  quantity  added  to  any  term  of  an 
arithmetical  progression  to  produce  the  next  term,  is  called 
the  common  difference. 

339.  If  the  common  difference  is  positive,  the  series,  or 
progression,  is  an  ascending  one ;  if  negative,  a  descending 
one. 

Thus,  «,  3  a,  5  a,  7  a,  etc.,  is  an  ascending  series ; 
and,     7  a,  5  a,  3  a,  a,  etc.,  is  a  descending  series. 

340.  In  the  general  discussion  of  arithmetical  progres- 
sions, a  represents  the  first  term,  d  the  common  differ- 
ence, I  the  last  term,  n  the  number  of  terms,  and  S  the 
sum  of  the  terms. 

341.  If  we  represent  the  first  term  by  a  and  the  com- 
mon difference  by  dy  the 

2d  term  =  a-\-d  4th  term  =  a  +  3  c? 

3d  term  =  a  -f-  2  c?  5th  term  =  a  +  4  <? 

Here  we  observe  that  each  term  equals  the  first  term 

plus  the  common  difference  multiplied  by  the  number  of 

terms  less  one  ;  hence,  the  nt\\  term  =  a  +  (^  —  1)  ^ ;  but 

the  »th  term  is  the  last  term  I ;  therefore, 

Prin.  132,    l  —  a-{-{n  —  l)d.     [Formula  A.] 


274  ELEMENTARY  ALGEBRA,  ^ 

342.  If  we  represent  the  sum  by  8,  we  have,     ^-2r-rw  =  J^- 
^=a  +  (a  +  ^)  +  («  +  ^^)+  ....         ^^-^'*^:rA/^ 
{l-%d)^{l-d)-\-l    (A) 

If  we  write  the  series  in  an  inverse  order, 

^S'  =  /  +  (Z  -  «f)  4-  (^  -  2  ^)  +  . . . . 

(a-\-%d)^{a-\-d)^a    (B) 
Add  (A)  and  (B), 

2^=(^  +  «)4-(Z  +  a)  +  (?  +  a)+.... 

(^  +  «^)  +  (^  +  «)  +  (^  +  «);   or 
2  /S'  =  (Z  +  ^)  taken  /^  times  =-{}-\-a)n\  therefore, 

rn 

THn.  133.     8=(l-\-a)'^.     [Formula  B.] 

lit 


Examples  involving  Arithmetical  Progressions. 

Illustrations. — 1.  Find  the  last  term  and  sum  of  the 
series  :  3,  7,  11,  15,  etc.,  to  10  terms. 

Solution :  Here  a  =  3,  c?  =  4,  and  n  =  10.    Substitute  these  values 
in  formula  A : 

l  =  a  -^  {n—l)d 
^  =  3  +  (10  -  1)  X  4  =  39 
Substitute  the  values  of  Z,  a,  and  n  in  formula  B : 

>S=(Z  +  a)| 

^=(39  +  3)  X  5  =  210 


2.  The  first  term  of  an  arithmetical  progression  is  25, 
the  number  of  terms  is  6,  and  the  sum  of  the  terms  is  102. 
Required  the  last  term. 

Solution  :  Substitute  the  values  a  =  25,  n  =  6,  and  yS'  =  102  in 
formula  B: 

102  =  (Z4-25)3  =  3Z  +  75 
dl  =  27 
1  =  9 


ARITHMETICAL  PROGRESSIONS.  275 

3.  Given  /  =  31,  ^  =  4,  and  >S'=  136,  to  find  w. 
Solution :  Substitute  these  values  in  formulas  (A)  and  (B) : 

1.  Since  l=za  +  {n—\)d,  31  =  a  +  (n— 1)4,  or  a  +  4?»  =  35     (A) 

2.  Since  >S'  =  (/  +  a)^,  136  =  (31  +  a)^,  or  31;i  +  an  =  272     (B) 

Transpose  (A),  a  =  35  —  4  n  (1) 

Substitute  (1)  in  (B), 

31w  +  35n-4n«  =  272  (2) 

Transpose  (5),  4  ;i2  _  66 » =  -  272  (3) 

Complete  the  square, 

^    9      «P      .    /33\'  1088      1089      1 

4n^-66n+  (^^j   =___  +  __  =  _        (4) 

Extract  the  V,  ^" ~  ^  ~  ^  ^  ^^^ 

Transpose,  2n  =  16  or  17 

Divide,  n  —  ^  or  8-^ 

Note. — Since  the  number  of  terms  is  a  whole  number,  8  is  the  true 
answer. 

EXERCISE     132. 

1.  Find  the  10th  term  and  the  sum  of  10  terms  of  the 
series  :  4,  8,  12,  etc. 

2.  Find  the  12th  term  and  the  sum  of  12  terms  of  the 
series  :  27,  25,  23,  etc. 

3.  Find  the  9th  term  and  the  sum  of  9  terms  of  the 

1,5,7       , 
series:  -+g+^,  etc. 

4.  Find  the  nth  term  and  the  sum  of  the  n  terms  of 
the  series :  1,  2,  3,  etc. 

6.  Find  the  rth  term  and  the  sum  of  r  terms  of  the 
series  :  2,  4,  6,  etc. 

6.  Given  a  =  3,  Z  =  28,  and  w  =  6,  find  d. 

7.  Given  >S'=112,  w  =  7,  and  a  =  25,  find  I  and  d, 

8.  Given  w  =  8,  a  =  8,  and  6?  =  5,  find  S  and  L 

9.  Given  (i=l-,  >S=58,  and  a  =  2,  find  I  and  n. 


276  ELEMENTARY  ALGEBRA. 

10.  Show  that  d  = ii.  Show  that  n  = 


l  +  a 


12.  Show  that  n  =  — -, — f-  1 

d 

13.  Show  that  1  = a      14.  Show  that  a  = 

n  n 

15.  Show  that  Z  =  —  +  ^  ~     x  d 

n         2 

16.  Show  that  /S=^w[2a  +  (?^  — 1)^] 


17.  Show  that  8 


2 

2     '^     2d 


18.  Show  that  a  = ^ —  d 

n  2 

19.  Show  that  d  =  -^ r^^ 

n  (n  —  1) 

20.  Giyen  d  =  -^,  ?  =  6-,  aud  8=  45,  to  find  a  and  w. 

21.  Given  (?  =  4,  >S'=  190,  and  a  =  1,  to  find  /  and  n, 

22.  Given  ^Z  =  3,  /  =  35,  and  /S  =  220,  to  find  n  and  a. 


23.  Show  that  n  =  ^i-^''±^(^'^-dr  +  SdS 

2d 


Concrete  Examples  involving  Arithmetical 
Progressions. 

Blustrations. — 1.  Insert  three  arithmetical  means  be- 
tween 3  and  11. 

Solution :  Since  there  are  to  be  three  arithmetical  means,  the  num- 
ber of  terms  is  5,  the  first  term  is  3,  and  the  last  term  is  11. 
Take  l=:a  +  (n-l)d 

Substitute,     11  =  d  +  4cd 
d  =  2 
,  • .    The  means  are  5,  7,  and  9. 


CONCRETE  EXAMPLES.  277 

2.  Find  the  series  whose  nth.  term  is  4  w  —  1, 

Solution :  Since  the  nth  term  may  be  any  term, 

Let  n  =  1,  then  nth  term  =  1st  term  =   4  —  1  =   3 
Let  n  =  2,  then  /ith  term  =  2d   term  =   8  —  1  =    7 
Let  n  =  3,  then  7tth  term  =  3d  term  =  12  —  1  =  11 
Etc.,  etc.,  etc. 


3.  Find  the  series  the  sum  of  n  terms  of  which  is  n^-\-  n. 

Solution : 

Let  /S'=n«  +  7i 

Let  w  =  1,  then  S  =  first  term  =  3 
Let  n  =  2,  then  S  =  sum  of  two  terms  =   4  +  2  =   6 
Let  n  =  3,  then  S  —  sum  of  three  terms  =   9  +  3  =  12 
Let  n  =  4,  then  S  =  sum  of  four  terms  =  IG  +  4  =  20 
Etc.,  etc.,  etc. 

Since  the  sum  of  two  terms  is  6  and  the  first  term  is  2,  the  second 
term  is  6  —  2  =  4. 

Since  the  sum  of  three  terms  is  12  and  the  sura  of  two  is  G,  the 
third  term  is  12  —  C  =  6. 

Similarly  the  fourth  term  is  20  —  12  =  8. 

And  the  series  is  2,  4,  G,  8,  etc. 


4.  The  sum  of  five  numbers  in  arithmetical  progression 
is  30,  and  the  difference  of  the  squares  of  the  extremes  is 
96.     Required  the  numbers. 

Solution :  Let  x  equal  the  middle  term  and  y  the  common  differ- 
ence, then  will  the  numbers  be 

x  —  2y,x  —  y,x,x-\-y,x-\-'Zy  (A) 

Since  their  sum  is  30,  5  a;  =  30  (1) 

a;=   6  (2) 

Since  the  difference  of  the  squares  of  the  extremes  is  96, 

(a;  +  2y)«-(x-23/)»  =  9G  (3) 

Expand  and  collect  terms,  8  x  y  =  9G  (4) 

a;y  =  12  (5) 

Substitute  (2)  in  (5)  and  reduce,      y  =   2 

Substitute  the  values  of  x  and  y  in  (A),  the  series  becomes  2,  4, 
6,  8,  10.  

5.  The  sum  of  four  numbers  in  arithmetical  progression 
is  24,  and  the  product  of  the  means  is  35.  Find  the  num- 
bers. 

13 


278  ELEMENTARY  ALGEBRA. 

Solution:  Let  x  —  y  and  x  +  y  h&  the  two  means,  the  common 
difference  being  2y,  then  will  the  series  be 

x  —  ^y,  x  —  y,  x-\-y,  x-\-^y  (A) 

Since  the  sum  is  24,  4  a;  =  24  (1) 

a:=    6  (2) 

Since  the  product  of  the  means  is  35,    x^  —  y^  =  35  (3) 

Substitute  (2)  in  (3)  and  reduce,  2/  =  ±  1         (4) 

Substitute  the  values  of  x  and  y  in  (A),  the  series  becomes 

8,  5,  7,  9, 

or    9,  7,  5,  3 


343.  Any  arithmetical  series  of  an  even  number  of  terms 
may  be  formed  by  putting  x^  y  and  x-\-y  for  the  two 
middle  terms,  making  2  y  the  common  difference. 

344.  Any  series  of  an  odd  number  of  terms  is  more 
conveniently  formed  by  putting  x  for  the  middle  term  and 
y  for  the  common  difference. 

EXERCISE     133. 

1.  The  sum  of  three  numbers  in  arithmetical  progres- 
sion is  30,  and  their  product  is  910.  Required  the  num- 
bers. 

2.  The  amounts  of  $100  for  1,  2,  and  3  years  respect- 
ively are  1105,  $110,  and  $115.  What  is  the  amount  of 
the  same  sum  for  15  years  ? 

3.  What  is  the  amount  of  $200  for  10  years  at  6^, 
simple  interest  ? 

4.  Insert  three  arithmetical  means  between  2  and  22. 

5.  Find  the  sum  of  all  the  whole  numbers  from  1  to 
100,  inclusive. 

6.  There  are  four  numbers  in  arithmetical  progression 
whose  sum  is  38,  and  the  product  of  the  extremes  is  70. 
Find  the  numbers. 

7.  If  the  nth.  term  of  a  series  is  2  ^  -—  1,  what  is  the 

series  ? 


CONCRETE  EXAMPLES,  279 

8.  If  a  body  falls  through  16yi2  feet  the  first  second, 
three  times  as  far  the  next  second,  five  times  as  far  the 

next,  and  so  on,  how  far  will  it  fall  in  half  a  minute  t  /  ^^y^^' 

9.  A  man  walks  1  mile  and  back  the  first  day,  2  miles 
and  back  the  second,  3  miles  and  back  the  third,  and  so 
on.     In  how  many  days  will  he  walk  72  miles  ? 

10.  A  man  put  out  at  interest  II  at  the  end  of  each 
month  for  10  years.  What  did  the  interest  amount  to  at 
6^  simple  interest? 

11.  The  sum  of  five  numbers  in  arithmetical  progression 
is  15,  and  the  sum  of  their  squares  is  55.  Find  the  num- 
bers. 

12.  If  the  sum  of  n  terms  of  a  series  is  ,  find 
the  series. 

13.  A  travels  2  miles  the  first  day,  4  the  second,  6  the 
third,  and  so  on.  Five  days  later  B  starts  out  and  travels 
uniformly  24  miles  a  day.  In  how  many  days  will  he 
overtake  A  ? 

14.  The  product  of  two  numbers  is  28,  and  the  product 
of  the  two  arithmetical  means  between  them  is  60.  Find 
the  numbers. 

16.  A  man  increased  his  capital  stock  $500  at  the  end 
of  each  year  for  10  years,  and  then  had  invested  16500. 
What  was  his  capital  at  first  ? 


Geometrical  Progressions. 

Definitions  and   Principles. 

345.  A  series  in  which  each  term  after  the  first  is  de- 
rived from  the  preceding  one  by  multiplying  it  by  a  con- 
stant quantity,  called  the  ration  is  a  geometrical  progres- 
sion. 


280  ELEMENTARY  ALGEBRA. 

346.  If  the  ratio  is  greater  than  one,  the  series  is  an 
ascending  one  ;  if  less  than  one,  a  descending  one. 

Thus,  a,  3  a,  9«,  27  a,  etc.,  is  an  ascending  series ; 
and,     27a,  da,  3a,  a,  etc.,  is  a  descending  series. 

347.  If  we  represent  the  first  term  by  a,  the  ratio  by  r, 
the  number  of  terms  by  n,  and  the  last  term  by  I,  the 

2d  term  =  ar  4th  term  =  ar^ 

3d  term  =  ar^  5th  term  =  ar^ 

Here  we  observe  that  each  term  equals  the  first  term 
multiplied  by  the  ratio  raised  to  a  power  whose  exponent 
is  one   less  than  the  number  of  terms ;  hence,  the  nth. 
term  =  a  r""^.     But  the  ^tlx  term  is  L     Therefore, 
Pfin.  134.     I  =  ar"~^     [Formula  A.] 

348.  If  we  represent  the  sum  of  a  geometrical  series  by 
S,  we  have, 

S=za  +  ar-\-ar^-\-ar^-\- ....-  +  1      (1) 
Multiply  (1)  by  r, 

j'S=ar-\-ar^-{-ar^-\-....l-{-lr  (2) 

Subtract  (1)  from  (2), 
(r-l)S=Ir-a,     Therefore,  (3) 

Prin.  135,     8=    ^Z\'     [Formula  B.] 
Cor.     S  = r— J  since  lr  =  ar''. 


Examples  in*  Geometrical  Progression. 

Illustrations. — 

1.  Find  the  9th  term  of  the  series  :  2,  4,  8,  etc. 
Solution  :    Here  a  =  2,  r  =  3,  and  n  =  ^.     Substitute  these  values 
in  formula  (A), 

Z  =  ar"-i  =r  2  X  28  =  29  =  512. 


EXAMPLES  IN  GEOMETRICAL  PROGRESSION,    281 

2.  Find  tlie  sum  of  7  terms  of  the  series  :  3,  9,  27,  etc. 
Solution  :  Here  a  =  3,  r  =  3,  and  n  =  7,  to  find  I  and  S, 
Substitute  in  formulas  (A)  and  (B), 

1.  Z  =^  ar*-i  =  3  X  3«  =  3' =  2187 


2.5  =  "-- 
r — 

.a^2187x  3-3^33^^ 

3.  Given  a  =  2,  r- 

=  2,  and  I  =  256,  find  n. 

Solution  :  Substitute  these  values  in  formula  (A), 

l  =  ar*-^ 

.'. 

256  =  2  x2«-i            (1) 

Divide  by  2, 

128  =  2—1,                   (2) 

But 

128  =  2',                       (3) 

.*. 

2«-i  =  2\                       (4) 

or 

n-l  =  7,                         (5) 

and 

n  =  8                         (6) 

4.  Given  I  =  320,  r  =  2,  and  n  =  7,  to  find  a  and  /SI 
Solution :  Substitute  these  values  in  formulas  (A)  and  (B), 

1.  l  =  ar^-^ 

320  =  ax2«  =  64a 
and  a  =  5 

2.  ^^^r-a^3a0.2-5^g3g 


EXERCISE     134. 

1.  Find  the  8th  term  of  the  series  :  2,  6,  18,  etc. 

2.  Find  the  7th  term  of  the  series  :  4,  —  12,  36,  etc. 

3.  Find  the  8th  term  of  the  series  :  162,  54,  18,  etc. 

4.  Find  the  10th  term  of  the  series  : 

^11         1       . 

5.  Find  the  nth  term  of  the  series  :  1,  2,  4,  etc. 

6.  Find  the  sum  of  6  terms  of  the  series  : 

3  +  12  +  48,  etc. 

7.  Find  the  sum  of  7  terms  of  the  series  :  1,  ^,  -,  etc. 

8.  Find  the  sum  of  n  terms  of  the  series  :  1,  — ,  j,  etc. 


282  ELEMENTARY  ALGEBRA. 

9.  Given  a  =  3,  r  =  2,  and  n  =  6,  find  I  and  S 

10.  Given  a  =  3,  r  =  3,  and  ;S'  =  363,  find  I  and  w 

11.  Given  /*  =  ^,  S=  r^r^,  and  7^  =  6,  find  I  and  a 

12.  Given  a  =  2  r,  r  =  r,  and  n  =  10,  find  ?  and  S 

13.  Show  that  a  =  -—r  14.  Show  that  /•  —  ""^Z 


'V^: 


15.  Show  that  S  = 

r*  —  r*~* 

16.  Show  that  a={l-S)r  +  S 

17.  Show  that  ^=^^^~^)  +  ^ 

r 

18.  Show  that  ar""  —  Sr  =  a  —  8 

19.  If  the  first  term  of  a  geometrical  progression  is  2, 
the  number  of  terms  4,  and  the  sum  of  the  terms  80, 
what  is  the  series  ? 

20.  Show  that  the  following  series  are  in  geometrical 
progression  : 

1.  7?,  xy,  y^  3.  -,  X,  y,  | 

2.  X,   ^fxy,  y  4.  x,  xy,  xy^,  xf 


Concrete  Examples  involving  Geometrical 
Progressions. 

Illustrations. — 1.    Insert   three   geometrical  means  be- 
tween 3  and  764. 

Solution  :  Since  there  are  to  be  three  means,  the  number  of  terms 
in  the  series  will  be  5,  the  first  term  3,  and  the  last  term  768. 
Take  Z  =  ar«-i 

Substitute,    768  =  3  r* 
r4  =  256 
r  =  4 
.*.    The  means  are  13,  48,  and  193. 


CONCRETE  EXAMPLES,  283 

2.  Find  the  series  whose  wth  term  is  2". 

Solution  :  Since  the  nth  term  may  be  any  term, 
Substitute  »  =  1,  n  =  2,  w  =  3,  etc.,  in 
nth  term  =  2" 
1st  term  =  2'  =    2 
2d  term   =  2«  =   4 
3d  term   =  2^  =   8 
4th  term  =24  =  10 
Etc.,  etc.,  etc. 


3.  Find  the  series  the  sum  of  n  terms  of  which  is  3"  — 1. 

Solution  :  Given  >S  =  3»  —  1, 

Let  n  =  1,  then  S  =  first  term  =  3  -  1  =   2 

Let  n  =  2,  then  S  =  sum  of  two  terms  =  3*  —  1  =  8 
Let  n  =  3,  then  S,  =  sum  of  three  terms  =  3^  —  1  =  26 
Let  n  =  4,  then  S  =  sum  of  four  terms  =  3*  —  1  =  80 

Since  the  sum  of  two  terms  is  8  and  the  first  term  is  2,  the  second 
terra  is  8  -  2  =  6. 

The  third    term  is  26  -   8  =  18. 

The  fourth  term  is  80  -  26  =  54. 

.-.    The  series  is  2,  6,  18,  54,  etc 


4.  The  sum  of  three  numbers  in  geometrical  progression 
is  63,  and  their  product  is  1728.     Find  the  numbers. 

Solution :  Let  x^,  xy^  and  y'  be  the  numbers, 

then  x^  +  xy  +  }/  =  QZ  (A) 

and    _  2^3^  =  1728  (B) 

Extract  the  Vb,  xy  =  n  (1) 

Add  (1)  to  (A),  a;«  +  22:y  +  2/'  =  75  (2) 

Extract  V(2),  a;  +  y  =  ±  5  ^/~^  (3) 

Subtract  3  times  (1)  from  (A), 

a;«_2a;y  +  3/2  =  27  (4) 

Extract  V(4),  a;  -  y  =  ±  3  \/3  (5) 

Add  (5)  and  (3),  2a;  =  ±  8  \/3  (6) 

a;=±4V3  (7) 
a;«  =  48 

Subtract  (5)  from  (3),  2y  =  ±  2  \/3  (8) 


The  numbers  are  48,  12,  and  3. 


y=±\/3 
y«  =  3 


284  ELEMENTARY  ALGEBRA, 

EXERCISE     133. 

1.  A  man  increases  his  capital  stock  at  the  end  of  each 
year  by  Vs  of  itself.  If  he  begins  with  $100,  what  will  his 
stock  be  at  the  end  of  8  years  ? 

2.  If  a  rangeman  begins  with  256  head  of  cattle,  and 
increases  his  herd  each  year  by  25^  of  itself,  in  how  many 
years  will  he  have  625  cattle  ? 

3.  A  has  twice  as  much  money  as  B,  B  twice  as  much 
as  0,  C  twice  as  much  as  D,  and  D  twice  as  much  as  E, 
and  they  together  have  16200.     How  much  has  each  ? 

4.  Insert  two  geometrical  means  between  Yg  and  Yig. 

5.  Insert  three  geometrical  means  between  2  Yg  and  *^y5i2. 

6.  An  elastic  ball  is  thrown  up  10  feet,  then  falls  and 
rebounds  5  feet,  then  falling  rebounds  2Y2  feet,  and  so  on. 
How  many  times  must  it  rebound  to  pass  over  38  Ys  feet  ? 

7.  Show  that  the  geometrical  mean  between  a  and  h  is 
Va/b.     Find  the  geometrical  mean  between  2  and  8. 

8.  There  are  three  numbers  in  geometrical  progression. 
The  product  of  the  first  two  is  75,  and  the  product  of  the 
last  two  is  225.     Find  the  numbers. 

9.  There  are  four  numbers  in  geometrical  progression. 
The  product  of  the  first  and  third  is  9,  and  the  product  of 
the  second  and  fourth  is  81.     Find  the  numbers. 

10.  The  sum  of  three  numbers  in  geometrical  progres- 
sion is  84,  and  the  quotient  of  the  third  and  first  is  16. 
Find  the  numbers. 

11.  Find  the  series  whose  nih  term  is  2  X  3*. 

12.  The  sum  of  three  numbers  in  geometrical  progres- 
sion is  42,  and  the  sum  of  their  squares  is  1092.  Find 
the  numbers. 

13.  The  three  digits  of  a  number  are  in  geometrical 
progression  ;  the  sum  of  the  digits  is  14  and  their  product 
is  64.     Find  the  number. 


INFINITE  SERIES.  285 

14.  The  sum  of  n  terms  of  a  series  is  ^2  (3"  —  1).  Find 
the  series. 

15.  A  milkman  drew  a  gallon  of  milk  from  a  can  con- 
taining 40  quarts,  then  put  in  the  can  a  gallon  of  water ; 
he  then  drew  off  a  gallon  of  the  mixture  and  put  in  its 
place  a  gallon  of  water.  He  did  this  five  times.  What 
part  of  the  contents  then  was  water  ? 

16.  Find  the  sum  of  n  terms  of  the  series  — ,  x,  y^  — , 

etc.  y        "^ 

17.  Insert  three  geometrical  means  between  3  and  243, 
and  find  their  sum. 


Infinite  Series. 


Definitions  and   Principles. 

349.  Any  series  of  an  unlimited  number  of  terms  is 
called  an  Infinite  series. 

350.  When  the  sum  of  n  terms  of  a  series  constantly 
approaches  some  definite  value  as  n  increases  indefinitely, 
the  series  is  said  to  be  convergent. 

Thus,  1  +  2"^4"^8'^16"^  ^^^''  ^^  ^  convergent 
series,  since  the  greater  the  number  of  terms  taken,  the 
nearer  will  their  sum  approach  to  2. 

351.  A  series  that  is  not  convergent  is  called  divergent 

352.  The  limit  of  a  convergent  series  is  the  value  whicli 
the  sum  of  n  terms  continually  approaches  as  w  is  in- 
creased indefinitely,  but  which  it  can  never  quite  reach, 
though  it  may  be  made  to  differ  from  it  by  less  than  any 
assignable  quantity. 

Thus,  2  is  the  limit  of  l-h^  +  T  +  3  +  ^+  etc. 

/«        4       o        lo 


286  ELEMENTARY  ALGEBRA. 

353.  In  any  geometrical  progression, 

ar""  —  a      a  —  af         a  af 


&■ 


r  —  \  1  —  r        1  —  r      1  —  r 

Suppose  r  <  1,  then  r*  approaches  0  as  a  limit  as  n  is 

increased  indefinitely,  and  hence  the  limit  of  z is  also 

0  ;  and  the  limit  (L)  of  ^S'  is • 

^    '  1  —  r 

Therefore, 

rHn.  136.     L  -  — ^  (0) 

1  —  r  ^  ' 


Examples  involving  Infinite  Series. 
Illustrations. — 
1.  Find  the  limit  of  the  series  :    3,  1,  -,  -,  etc. 

Solution:   Here  a  =  3,  and  »*  =  -q-.     Substitute  these  values  in 
formula  (C),  "^ 

J         «  3  1 

3 
,  2.  Find  the  value  of  the  circulating  decimal  -36. 

solution :  -36  =  -363636,  etc.  =  ^  +  j||g  +  ^^  +  etc. 

36  1 

Here  a  =  ^kk  and  r  =  j^.    Substitute  these  values  in  (C), 


L  = 


36^ 

100  36       4 


1-r       .  _  J_       99      11 
100 


3.  Find  the  value  of  the  circulating  decimal  '24. 
solution  :  .24  =  -24444,  etc.  =  ^^^(^^^^^^^  etc.  ) 

rpu    T    V    4^  /  4     ,      4      .      4  ,    \  a  100  4 

The  limit  of  (^— +  _  +  j^Q+  etc.j=^^^  =  — ^  =  ^ 


10 
10  "^  90  "~  90  "^  90  ~  90  ~  45 


EXAMPLES  INVOLVING  INFINITE  SERIES.      287 

4.  A  hound  is  20  rods  behind  a  fox,  and  runs  2  rods 
to  the  fox's  one.  Ilow  far  must  the  hound  run  to  catch 
the  fox  ? 

Solution  :  While  the  hound  runs  the  20  rods  the  fox  is  ahead,  the 
fox  runs  10  rods ;  then,  while  the  hound  runs  these  10  rods,  the  fox 
runs  5  rods ;  then,  while  the  hound  runs  these  5  rods,  the  fox  runs 
2  Vx  rods,  etc.    Therefore  the  hound  runs  in  all  the  sum  of 

20  rd.  +  10  rd.  +  5  rd.  +  2  g-  rd.  +  etc. ;  or,  since 

^         a            20  ._      , 

L  =  z = =  40  rods. 

2 


EXERCISE     136. 

1.  Find  the  limit  of  the  series  : 

1.  2  +  l+l  +  etc.  3.  10  +  1  +  1  +  etc. 

2.  9  4-3  +  1  +  etc.  4.  12^  + 6^  +  3^  +  etc. 

/*  4  o 

2.  Find  the  limit  of  the  series  : 

1.  a  +  ^  +  ^  +  etc.  3.  a-^b  +  -  +  etc. 

or  3j       cti        cty 

2.  ax-{-x-\ f-  etc.         4.  -  +  -« +  -^  +  etc. 

3.  Find  the  value  of  : 

1.  -45  3.  i728  5.   -012 

2.  -124  4.   -36  6.   -012 

4.  A  ball  is  thrown  up  10  feet,  falls  and  rebounds  5 
feet,  then  falls  and  rebounds  2  72  feet,  and  so  on.  How 
far  does  it  move  before  it  stops  ? 

5.  An  oflScer  is  100  rods  behind  a  thief,  and  goes  3  rods 
while  the  thief  goes  2  rods.  How  far  must  the  oflBcer  go 
to  catch  the  thief  ? 

6.  At  what  time  after  4  o'clock  are  the  hour  and  minute 
hands  of  a  watch  together  ? 


CHAPTER  VIII. 


Miscellaneous  Examples. 

EXERCISE     137. 

1.  It  x  =  l,  y  =  d,  Z  =  6,   U  =  0, 

find  the  value  ot  x^ -^  2  y^ -\- 3  z^ -\- 4t  u- 

2.  If  a;  =  2,  y  =  0,  z=i-l,  u  =  l, 

find  the  value  of  {xy  —  uz){yz  —  ux){uy  ^  xz) 

3.  Find  the  value  of  : 

2x^-^2f-2z^  +  ^xy  1 

Zx'-'dy'-Zz^^Qyz  '^  ^  ~  *'  ^"2'      ^^ 

4.  Add  3«^+4^+ g^.    4^""3^  +  i^'  ^^^ 

6^-4^  +  3^ 

5.  Sax-{-6I?x  —  2cx  +  7dx  —  4:ax-\-6dx-{- 

llcx  ~  6clx  equals  how  many  times  x  ? 

.   T?         3         4,2^,1,3         4 

6.  From  -x--y-i--z  take  _:z:  +  -^--2; 

7.  Simplify  a— [{  —  a  —  {a-{- a)  —  a]  —  a  — (a-\- a)] 

8.  From  o^m^  +  ^mTe  +  cw^  take 

(b  —  c)  m^  —  (a  —  c)mn  —  (b  —  a)  n^ 

9.  What  is  the  value  of  4:(mq  —  np)  —  {{m  —  n)  — 

{p-q)V,   if  ??i  =  0,  /^  =  2,  ^  =  —  3,  (7  =  4? 

10.  Multiply  x^  -\-x^  y^  -\-  y^  by  x~i  +  ^~^  y~^  +  2/~^ 


11. 


Expand  (.i  +  i)(.i-i)(.+  l)(.^  +  fg) 


MISCELLANEOUS  EXAMPLES. 


289 


12.  Divide  x^-\-  (a-\-h-\-c) xr-\-  {ab-\- ac-\-  bc)x-{-ahc 

hy  x-\-b 

13.  Write  the  quotient  of  a}""  +  b'""  divided  by  a^  +  b- 

14.  Factor  x^- -\- y^^  16.  Factor  x^ -\- x!^  y^ -\- y^ 

16.  Find  the  H.  C.  D.  of  ax-\-ay -{-bx-\-by  and 

cx-\-cy  —  dx  —  dy 

17.  Find  the  L.  C.  M.  of  9  a;^  -^  12  a:  +  4  and 

Zx^^llx-\-(j 


18.  Simplify  -^ 


^-\-^y-{-y^  ^^-{-y^ 


X 


xy-\-y^      x'  —  f 


19.  Reduce 

>^-y^^2yz-z^ 

;  to  its  lowest  terms. 

20.  Simplify  ^  +  ^     P-^        ^,M, 

'  p-q    p^-q    q^-p"" 

21.  Simplify : 

qr                         pr                        pq 

(p-q){p-r)       {q-p){r-q)      (r-q)(r-p) 

22.  Simplij 

1              a 

«  + 1       a^  —  1 

''    1 

1  -  fl2      i-^a 

23.  Solve 

*^  +  4,7.-3      ^^^  +  ^1,    9 

7         '  2  +  (ix~        7         '  28 

i      x-\-y-z  =  3) 
24.  Solve  <      x  —  y-^z  =  5> 

i-x-\-y-^z  =  7)      . 

[      1  +  1-1  =  1^ 

26.  Solve  ■ 

1-1+1=3 

X      y      z 

-1+1+1=5 

26.  Square  Vz-\-  V^  —  Vz      27.  Cube  1  ~  V2 


290  ELEMENTARY  ALGEBRA. 

28.  Cube  a-\-Vax-{-x  29.  (x-^-)   ^ -^  (a;^') »  =  ? 

30.  Expand  (^^"^  +  -^13)  and  express  the  result  with- 
out negative  exponents. 

31.  Multiply  2  Vis  +  2  V288  -  3  V32  -  VT28  by  a/2 

32.  Simplify  {x  +  V^^)^  {x  —  V—yY 

33.  Solve  a^-\-{a-\-b-\-c)x=  —i{a-^c) 


34.  Solve     ^    , =  =  ^ 

a;  —  V  2  —  a;2      3 


35.  Solve  2x^  +  dV2x^-\-3x  =  18-3x 

36.  Solve   i  a   o      o  o    2      00  r 

37.  Extract  the  square  root  of  : 

x  —  2x^y^-\-3x^y^  —  2x^i/^-\-y 

38.  Extract  the  cube  root  of  : 

a^-dx^+6x^-7x-{-6x^-3x^-\-l 

39.  Divide  x!^ -{- a^ y^ -\- y^  by  x-{-  Vxy  +  y 

40.  Simplify  : 

1  1  . 


x(x—y){x-z)      y{y-x){z-y)      z{x-z){y-z) 

41.  Find  the  value  of  7?  —  xy  -\-y^  when 

a  —  l)       T  a-\-'b 

X  =  — —7  and  V  = 7 

a-{-b  ^      a  —  h 

42.  Multiply  ma^-{-nx  —p    by    ax  —  b,   and   inclose 
the  coefficients  of  the  different  powers  of  x  in  parentheses. 

43.  Factor  ^-\  and  a^  ^3aH  -  4.al)^ -121)^ 

44.  Put  y  for  a;  +  -  in  the  following  expressions,  and 

X 

simplify:  ^'  +  -3;  ^'  +  ^5  ^  +  ^ 


MISCELLANEOUS  EXAMPLES.  291 

45.  Factor  32  3^-]-z^  and  x^  +  xt/-\-y^ 

46.  Raise  1  —  V—  3  to  the  fourth  power. 

47.  Place  the  monomial  factors  of  the  following  expres- 
sions within  the  parentheses  : 

48.  Simplify  1  -  [-  1  -  {-  1  -  (-  1)  -  Ij  -  1]  -  1 

49.  Simplify  [](-2)-2|-2]~' 

50.  Solve  x-^-\ :  =  x-^ 2 

x-^  x~^ 

61.  Solve  x  = y  and  y  = x 

X     ^  ^      y 

^  ,       a:  +  2      x  —  %      rr  +  4      x  —  4: 

52.  Solve  — hs -o  =  — —^ n 

x-\-Z      x  —  d      x^5      x-^6 

53.  Develop      _      into  a  series  by  division.     By  the 

law  of  the  series,  what  will  the  10th  term  be  ?    What  the 
wth  term  ? 

64.  Simplify  and  clear  of  negative  exponents  : 

^l!jfr:!.  ^iHjr^.  i^-y^)-^ 

«"*  — y~*'  x-^-\-y-^'   {x  -yy 
56.  Multiply  2  a/SO  -  Vis  +  Vll  by  a/S  +  V^  -  V^ 

56.  Rationalize  the  denominators  of  ; 

57.  Solve  -7=-' —  =  —7^ — 

Va:-|-6       Va;  +  9 

68.  Find  the  H.  C.  D.  of  2ax-{-%hx-\-l a  +  lh 

and  2cx  —  2dx-\-lc  —  ld 

69.  From  my^-\-nyz-\-rz^ 

take  (;i  —  r)y^  —  {m  —  r)yz  —  {n  —  m) z^ 
60.  Find  the  value  of  a;  in  a;:a  —  1::1  —  Va  :  1  +  Va 


292  ELEMENTARY  ALGEBRA, 


61.  Find  the  value  of  ^ 


X  —  y 

when  X  =  V2  and  y  =  —  a/2 

62.  Find  the  yalue  of  : 

1 -\- X -\- x^  -{- x^  -\-  etc.,  ad  infinitum,  when  x  =  - 

63.  Find  the  equation  whose  roots  are 

1  +  a/^  and  1  -  a/^ 

64.  Find  the  equation  whose  roots  are  a,  l,  c,  and  d, 

65.  Solve  ?>ax^ -\-%'bx^^c 

66.  Solve  ic  —  ^  x  =  a-\-  Va 


67.  Solve   J-  +  ^+V.  +  ^=     ^0 


^'r+       ^^      =3660) 

68.  If  x  =  y^-{-y-{- 1,  what  is  the  value  of  x^-\-x-{-l? 

69.  Substitute    ay^  —  hy   for   ic  in   x^  -{- x  y  -\r  y^ ,  and 
bracket  the  coefficients  of  the  like  powers  of  x. 


70.  Solve  /  „  = 7  by  proportion. 

71.  Solve  —J- T=  =  a—  ^Ta 

V  X—  V  a 

{  x^-{-  xy  -^y^  =  21  ) 

72.  Solve  ]     ;   .\  ;  *^      ^  [ 

i  x  -i-y  =35) 

^  .        {x4-xy4-xy^  =  26) 

74.  Solve    -^      '      ^3^    ^  ^^        ^ 

i  X  —  xy^  =  —  62        ) 


75.  Solve 


j  x^ -\- y^ -\- X  y  =  4t9 


49      I 
=  31  ( 


\xy-\-2x-\-2y 

76.  If  ^i  is  integral,  what  kind  of  number,  odd  or  even, 
is  represented  by  2^  +  1?    2^^  —  1?    2n? 

77.  Find        V  +  1,        V  — 1,     V+l,     vl6 
78..  Solve  "'"V^  =  -  1,    V^  =  -  1 


MISCELLANEOUS  EXAMPLES,  293 

EXERCISE     138. 

1.  John  and  James  together  had  $6800.  John  spent 
Vs  of  his  money,  and  James  Yi  of  his,  and  each  had  the 
same  sum  remaining.     How  much  had  each  at  first  ? 

2.  John  is  V3  as  old  as  his  father,  but  in  20  years  he 
will  be  Yi3  as  old.     How  old  is  each  ? 

3.  Divide  100  into  two  such  parts  that  the  quotient  of 
the  smaller  part  divided  by  the  difference  between  the 
parts  may  be  12. 

4.  The  sum  of  the  two  digits  of  a  number  is  10,  and  if 
36  be  added  to  the  number  the  order  of  the  digits  will  be 
reversed.     Find  the  number. 

5.  Find  a  fraction  such  that  if  2  be  added  to  the 
numerator  the  fraction  equals  1,  but  if  5  be  added  to  the 
denominator  the  fraction  equals  Yg. 

6.  If  a  rectangle  had  its  width  increased  by  4  feet  and 
its  length  diminished  by  8  feet,  it  would  become  a  square 
inclosing  an  equal  area.  Find  the  dimensions  of  the 
rectangle. 

7.  A  can  do  a  piece  of  work  in  2  hours  45  minutes,  and 
B  can  do  it  in  3  hours  15  minutes.  In  what  time  can  they 
do  it  working  together  ? 

8.  A  man  was  engaged  to  work  for  40  days  on  condition 
that  for  every  day  he  worked  he  was  to  receive  $3,  and  for 
every  day  he  was  idle  to  forfeit  $1Y2-  At  the  end  of  the 
time  he  received  $75.     How  many  days  was  he  idle  ? 

9.  At  an  election  there  were  three  candidates  for  sheriff. 
The  whole  number  of  votes  polled  was  5325.  B  received 
662  votes  more  than  C,  and  A's  majority  over  B  and  C 
was  1  vote.     How  many  votes  had  each  ? 

10.  One  man  rides  a  mile  on  a  bicycle  in  5  Ye  minutes ; 
another,  a  mile  in  6^3  minutes.  If  they  start  at  the  same 
time  from  two  towns  18  miles  apart  and  approach  each 
other,  in  what  time  will  they  meet  ? 


294  ELEMENTARY  ALGEBRA. 

11.  A  man  can  row  4  miles  an  hour  in  still  water,  and 
11  miles  down  a  river  and  back  again  in  6%  hours.  What 
is  the  velocity  of  the  current  ? 

12.  At  what  time  between  5  and  6  o'clock  are  the  hour 
and  minute  hands  of  a  watch  together  ?  At  right  angles  ? 
Opposite  each  other  ? 

13.  Find  two  numbers  in  the  ratio  of  2  to  3,  such  that 
if  each  be  diminished  by  12,  they  shall  be  in  the  ratio  of 
1  to  2. 

14.  A  boat  steaming  Yg  a  mile  an  hour  above  its  ordi- 
nary rate  gains  17  Y7  minutes  in  going  60  miles.  What  is 
its  usual  rate  ? 

15.  Six  silver  and  4  gold  pieces  are  worth  as  much  as 
16  silver  and  2  gold  pieces,  and  10  of  each  are  together 
worth  130.  What  is  the  value  of  a  gold  piece  ?  What  of 
a  silver  piece  ? 

16.  Two  numbers  are  in  the  ratio  of  p  to  q,  and  the 
sum  of  their  squares  is  jt?*  —  q^.     What  are  the  numbers  ? 

17.  Show  that  any  square  exceeds  a  rectangle  of  equal 
perimeter  by  the  square  of  Y2  the  difference  of  the  length 
and  breadth  of  the  rectangle. 

18.  A  man  bought  12  apples.  Had  he  bought  3  less 
for  the  same  sum,  they  would  have  cost  him  1  cent  apiece 
more.     What  did  he  pay  apiece  ? 

19.  A  and  B  together  have  110  sheep,  A  and  0  together 
100  sheep,  and  B  and  0  together  90  sheep.  How  many 
has  each  ? 

20.  Twelve  men  agreed  to  do  a  piece  of  work  in  a  given 
time,  but  4  men  did  not  report  for  work,  in  consequence 
of  which  the  time  had  to  be  extended  4  days.  What  was 
the  time  agreed  upon  ? 

21.  Two  trains  pass  a  station  at  an  interval  of  3  hours, 
moving  respectively  at  the  rate  of  20  and  32  miles  an  hour. 
In  what  time  will  the  fast  train  overtake  the  slow  train  ? 


MISCELLANEOUS  EXAMPLES,  295 

22.  What  principal  will  in  8  years  at  6^  produce  as 
much  interest  as  1800  in  9  years  at  8^  ? 

23.  The  population  of  a  Western  city  increases  annually 
10^  ;  it  is  now  29,282.     What  was  it  4  years  ago  ? 

24.  The  sum  of  the  fourth  powers  of  three  consecutive 
numbers  is  962.     Find  the  numbers. 

25.  A  number  of  two  digits  is  to  the  number  formed 
by  interchanging  the  digits  as  4  to  7,  and  the  difference  of 
the  two  numbers  is  27.     What  is  the  number  ? 

26.  A  man  saved  $1026  in  3  years.  How  much  were 
his  annual  savings  if  they  increased  in  geometrical  progres- 
sion, and  he  saved  $18  the  first  year  ? 

27.  The  sum  of  7  numbers  in  arithmetical  progression 
is  56,  and  the  sum  of  their  squares  is  560.  Required  the 
numbers. 

28.  The  volumes  of  two  stones  are  to  each  other  as 
3  to  4,  and  the  weights  of  equal  volumes  as  iVg  to  1. 
What  is  the  weight  of  each  if  their  united  weight  is  340 
pounds  ? 

29.  The  diagonal  of  a  rectangle  is  10,  but,  if  the  length 
be  increased  by  4  and  the  width  by  3,  the  diagonal  will  be 
15.     Required  the  length  and  width  of  the  rectangle. 

30.  A  man  has  three  horses.  The  value  of  the  first  is 
160,  the  value  of  the  second  equals  the  value  of  the  first 
and  Vs  the  value  of  the  third,  and  the  value  of  the  third 
equals  the  value  of  the  first  two.  Required  the  value  of 
the  three  together. 

31.  A  man  owns  $15,000  worth  of  stock,  part  of  which 
is  5  j^  and  part  6  ^  stocks ;  his  annual  income  is  $830. 
How  much  of  each  kind  has  he  ? 

32.  I  sold  a  horse  for  as  many  per  cent  above  $150  as  I 
lost  per  cent  on  the  cost,  and  lost  $45.  What  was  the  cost 
and  loss  per  cent  ? 


296  ELEMENT AR Y  ALGEBRA. 

33.  A  and  B  can  do  a  piece  of  work  in  5  days,  working 
10  hours  a  day ;  A  and  0  in  6  days,  working  9  hours  a 
day ;  and  B  and  0  in  8  days,  working  8  hours  a  day.  In 
what  time  can  each  alone  do  it,  working  8  hours  a  day  ? 

34.  A  is  50  yards  in  advance  of  B,  and  goes  1  yard  the 
first  minute,  3  yards  the  second,  5  yards  the  third,  and  so 
on ;  B  goes  uniformly  15  yards  a  minute.  In  how  many 
minutes  will  he  overtake  A  ? 

35.  A  passenger-train,  after  running  one  hour,  was  par- 
tially disabled,  and  could  run  only  at  Vg  of  its  usual  rate, 
which  caused  it  to  be  one  hour  late  at  its  destination.  Had 
the  accident  occurred  15  miles  farther  on,  the  train  would 
have  been  only  52  Yg  minutes  late.  What  was  the  usual 
rate  of  the  train  ? 

36.  The  sum  of  the  two  means  of  a  geometrical  progres- 
sion of  four  terms  is  36,  and  the  sum  of  the  extremes  is 
84.     Find  the  series. 

37.  The  product  of  five  numbers  in  arithmetical  progres- 
sion is  23,040,  and  their  sum  is  40.     Find  them. 

38.  The  sum,  the  product,  and  the  difference  of  the 
squares  of  two  numbers  are  equal.     Find  the  numbers. 

39.  A  is  a  feet  behind  B,  and  goes  m  feet  in  t  seconds, 
while  B  goes  n  feet  in  p  seconds.  In  how  many  seconds 
will  A  overtake  B  ? 

40.  If  from  a  number  whose  four  figures  are  in  arith- 
metical progression,  be  subtracted  the  number  formed  by 
reversing  the  order  of  the  digits,  the  remainder  will  be 
6174  ;  the  sum  of  the  digits  is  24.     Required  the  number. 

41.  Two  numbers  are  in  the  ratio  of  2  to  3,  and  if  2  be 
added  to  each  of  them,  they  will  be  in  the  ratio  of  3  to  4. 
Find  them. 

42.  A's  age  2  years  hence  is  to  his  age  3  years  ago  as 
9  times  his  age  3  years  ago  is  to  4  times  his  age  2  years 
hence.     Required  his  age. 


MISCELLANEOUS  EXAMPLES.  297 

43.  The  capacities  of  two  cubical  cisterns  are  to  each 
other  as  1  to  8  ;  but  if  2  feet  were  added  to  each  of  their 
dimensions,  their  capacities  would  be  to  each  other  as  27 
to  125.     Required  the  capacity  of  each. 

44.  The  sum  of  the  squares  of  three  numbers  is  29,  the 
sum  of  the  products  of  them  taken  two  together  is  26,  and 
the  first  is  5  less  than  the  sum  of  the  other  two.  Find 
the  numbers. 

45.  A  general  drew  up  his  army  in  the  form  of  a  square 
and  found  he  had  615  men  over ;  he  then  increased  the 
side  of  the  square  by  5  men,  and  lacked  60  men  to  com- 
plete the  square.     How  many  men  were  in  the  army  ? 

46.  A  man  has  a  farm  of  150  acres,  in  the  form  of  a 
rectangle,  whose  length  is  to  its  breadth  as  5  to  3.  A 
road  of  uniform  width,  containing  3^Y4o  acres,  surrounds 
the  farm,  and  is  a  part  of  it.     How  wide  is  the  road  ? 

47.  The  fore-wheel  of  a  carriage  makes  88  revolutions 
more  in  going  a  mile  than  the  hind-wheel,  but  if  the  cir- 
cumference of  i\i6  fore-wheel  were  diminished  2  feet  the 
fore-wheel  would  make  220  revolutions  more  than  the 
hind-wheel.     Required  the  circumference  of  each  wheel. 

48.  A  merchant  gains  annually  20  per  cent  of  his  capi- 
tal ;  of  this  he  spends  $1000,  and  adds  the  balance  to  his 
capital  for  the  next  year ;  at  the  end  of  4  years  his  stock 
is  $25,736.     What  was  his  original  stock  ? 

49.  A  man  starts  at  the  foot  of  a  mountain  to  walk  to 
its  top.  During  the  first  half  of  the  distance  he  walks 
Yg  a  mile  an  hour  faster  than  during  the  last  half,  and  he 
reaches  the  top  in  4  hours  24  minutes.  Returning,  he 
walks  Y2  a  mile  an  hour  faster  than  during  the  latter  half 
of  his  ascent,  and  completes  the  descent  in  4  hours.  Find 
the  distance  to  the  top  of  the  mountain. 

50.  A  lump  of  gold  22  carats  fine  contains  36  ounces 
of  alloy.  How  many  ounces  of  alloy  in  a  lump  of  the 
same  weight  only  16  carats  fine  ? 


298  ELEMENTARY  ALOEBRA. 

51.  In  a  mile  walk,  A  gives  B  a  start  of  1  minute  and 
overtakes  him  at  the  mile-post.  In  a  second  trial,  A  gives 
B  a  start  of  60  yards,  and  beats  him  10  seconds.  At  the 
rate  of  how  many  miles  an  hour  does  each  walk  ? 

62.  If  the  cost  of  an  article  had  been  8  ^  less,  the  gain 
would  have  been  10  ^  more.     Find  the  gain  per  cent. 

53.  A  railway-train,  after  traveling  for  1  hour,  has  an 
accident  which  delays  it  60  minutes,  after  which  it  pro- 
ceeds at  %  of  its  former  speed,  and  arrives  at  its  desti- 
nation 3  hours  behind  time.  Now,  had  the  accident 
occurred  50  miles  farther  on,  the  train  would  have  arrived 
1  %  hour  sooner.     What  is  the  length  of  the  line  ? 

54.  Two  men,  A  and  B,  engaged  to  work  for  a  certain 
number  of  days  at  different  rates.     At  the  end  of  the  time, 

A,  who  had  been  idle  4  days,  received  75  shillings ;  but 

B,  who  had  been  idle  7  days,  received  only  48  shillings. 
Now,  had  B  been  idle  only  4  days,  and  A  7  days,  they 
would  have  received  the  same  sum.  For  how  many  days 
were  they  engaged  ? 

55.  Three  pipes.  A,  B,  and  0,  can  fill  a  cistern  in  one 
hour.  B  delivers  twice  as  much  water  per  minute  as  A. 
0  alone  will  fill  it  in  one  hour  less  than  B  alone.  How 
long  will  it  take  each  to  fill  it  ? 


General  Definitions. 

1.  Quantity  is  anything  that  may  be  increased,  dimin- 
ished, and  measured. 

2.  Quantity  is  estimated  by  assuming  some  definite  por- 
tion of  it  as  a  standard  of  measure,  and  finding  how  many 
times  it  contains  this  standard. 

3.  Any  definite  portion  of  quantity  assumed  as  a  stand- 
ard of  measure  is  a  unit, 

4.  Number  is  that  which  denotes  how  many  units  a 
quantity  contains. 


GENERAL  DEFINITIONS,  299 

6.  A  quantity  that  contains  a  definite  number  of  units 
is  a  specific  quantity  ;  as,  five  pounds. 

6.  A  quantity  that  may  contain  any  number  of  units  is 
a  general  quantity  ;  as,  a  fiock. 

7.  The  number  of  units  in  a  specific  quantity  is  ex- 
pressed by  one  or  more  of  the  figures  of  arithmetic. 

8.  The  number  of  units  in  a  general  quantity  is  ex- 
pressed by  one  or  more  of  the  letters  of  the  alphabet,  or  by 
both  figures  and  letters. 

9.  By  a  figure  of  speech,  the  representation  of  the  num- 
ber of  units  in  a  quantity,  by  figures  or  letters,  is  also 
called  a  quantity. 

10.  When  the  number  of  units  in  a  quantity  is  denoted 
by  figures,  the  expression  is  called  a  numerical  quantity. 

11.  When  the  number  of  units  in  a  quantity  is  repre- 
sented wholly  or  partially  by  letters,  the  expression  is 
called  a  literal  quantity. 

12.  Quantities  which  are  opposed  to  each  other  in  char- 
acter—  that  is,  which  tend  to  destroy  each  other  when 
combined — are  positive  and  negative  quantities. 

13.  Of  two  opposite  quantities,  it  does  not  matter  which 
is  considered  positive  and  which  negative,  if  consistency  is 
maintained  throughout  the  operation  or  investigation  into 
which  they  enter. 

14.  The  number  of  units  in  a  positive  quantity  is  char- 
acterized by  placing  before  it  the  symbol  +  (p^^s) ;  and 
the  number  of  units  in  a  negative  quantity,  by  placing 
before  it  the  symbol  —  (minus).  This  peculiar  notation 
gives  rise  to  symbolized  numbers. 

16.  Arithmetic  is  the  science  of  numbers,  irrespective 
of  their  character  as  positive  or  negative.  Arithmetic  based 
on  the  literal  notation  is  Literal  Arithmetic. 

16.  Algebra  is  the  science  of  symbolized  numbers  as  the 
representatives  of  positive  and  negative  quantities. 


300  ELEMENTARY  ALGEBRA. 

Principles. 

1.  The  algebraic  sura  of  two  or  more  similar  terms  with 
like  signs  equals  their  arithmetical  sum  with  the  same 
sign  (page  16). 

2.  The  algebraic  sum  of  two  similar  terras  with  unlike 
signs  equals  their  arithmetical  difference  with  the  sign  of 
the  greater  (page  16). 

3.  The  algebraic  sum  of  two  or  more  dissimilar  terms 
equals  a  polynomial  composed  of  those  terms  (page  17). 

4.  The  algebraic  difference  of  two  quantities  equals  the 
algebraic  sum  obtained  by  adding  to  the  minuend  the  sub- 
trahend with  the  sign  changed  (page  23). 

5.  The  product  of  two  quantities  with  like  signs  is 
positive  (page  27). 

6.  The  product  of  two  quantities  with  unlike  signs  is 
negative  (page  27). 

7.  The  exponent  of  a  factor  in  the  product  equals  the 
sura  of  its  exponents  in  the  multiplicand  and  multiplier 
(page  28). 

8.  Multiplying  one  factor  of  a  quantity  multiplies  the 
quantity  (page  28). 

9.  Multiplying  every  term  of  a  quantity  multiplies  the 
quantity  (page  31). 

10.  The  quotient  of  two  quantities  with  like  signs  is 
positive  (page  33). 

11.  The  quotient  of  two  quantities  with  unlike  signs  is 
negative  (page  33). 

12.  The  exponent  of  a  factor  in  the  quotient  equals  the 
difference  of  the  exponents  of  the  factor  in  the  dividend 
and  divisor  (page  34). 

13.  Any  quantity  with  an  exponent  of  zero  equals  unity 
(page  34). 

14.  Dividing  one  factor  of  a  quantity  divides  the  quan- 
tity (page  35). 


PRINCIPLES.  301 

15.  Dividing  every  term  of  a  quantity  divides  the  quan- 
tity (page  37). 

16.  If  the  same  quantity  or  equal  quantities  be  added  to 
equal  quantities,  the  results  will  be  equal  (page  39). 

17.  If  the  same  or  equal  quantities  be  subtracted  from 
equal  quantities,  the  results  will  be  equal  (page  39). 

18.  If  equal  quantities  be  multiplied  by  the  same  or 
equal  quantities,  the  results  will  be  equal  (page  39). 

19.  If  equal  quantities  be  divided  by  the  same  quantity 
or  equal  quantities,  the  results  will  be  equal  (page  40). 

20.  A  term  may  be  taken  from  one  member  of  an  equa- 
tion to  the  other,  if  its  sign  be  changed  (page  40). 

21.  If  both  members  of  a  fractional  equation  be  multi- 
plied by  a  common  denominator  of  its  terms,  it  will  be 
cleared  of  fractions  (page  40). 

22.  If  the  sign  of  every  term  of  an  equation  be  changed, 
the  members  will  still  be  equal  (page  41). 

23.  If  a  number  of  terms  are  inclosed  by  a  parenthesis 
preceded  by  plus,  the  symbol  and  the  sign  before  it  may 
be  removed  without  altering  the  value  of  the  expression 
(page  50). 

24.  If  a  number  of  terms  are  inclosed  by  a  parenthesis 
preceded  by  minus,  the  symbol  and  the  sign  before  it  may 
be  removed,  if  the  sign  of  every  term  inclosed  be  changed 
(page  51). 

25.  Any  number  of  terms  may  be  inclosed  by  a  paren- 
thesis and  preceded  by  plus,  without  changing  the  value 
of  the  expression  (page  51). 

26.  Any  number  of  terms  may  be  inclosed  by  a  paren- 
thesis and  preceded  by  minus,  if  the  sign  of  every  term 
inclosed  be  changed  (page  51). 

27.  An  even  power  of  a  positive  or  a  negative  quantity 
is  positive  (page  62). 

28.  An  odd  power  of  a  quantity  has  the  same  sign  as 
the  quantity  (page  63). 

14 


302  ELEMENTARY  ALGEBRA. 

29.  Multiplying  the  exponent  of  a  factor  by  the  ex- 
ponent of  a  power  raises  the  factor  to  that  power  (page  63). 

30.  Raising  every  factor  of  a  quantity  to  a  given  power 
raises  the  quantity  to  that  power  (page  63). 

31.  The  square  of  the  sum  of  two  quantities  equals  the 
square  of  the  first,  plus  twice  their  product,  plus  the  square 
of  the  second  (page  65). 

32.  The  square  of  the  difference  of  two  quantities  equals 
the  square  of  the  first,  minus  twice  their  product,  plus  the 
square  of  the  second  (page  65). 

33.  The  cube  of  the  sum  of  two  quantities  equals  the 
cube  of  the  first,  plus  three  times  the  square  of  the  first 
into  the  second,  plus  three  times  the  first  into  the  square 
of  the  second,  plus  the  cube  of  the  second  (page  66). 

34.  The  cube  of  the  difference  of  two  quantities  equals 
the  cube  of  the  first,  minus  three  times  the  square  of  the 
first  into  the  second,  plus  three  times  the  first  into  the 
square  of  the  second,  minus  the  cube  of  the  second  (page 

35.  The  product  of  any  even  number  of  factors  with 
like  signs  is  positive  (page  67). 

36.  The  product  of  any  odd  number  of  factors  with 
like  signs  has  the  same  sign  as  the  factors  (page  67). 

37.  If  the  signs  of  an  even  number  of  factors  be  changed, 
the  sign  of  their  product  will  remain  unchanged  (page  67). 

38.  If  the  signs  of  an  odd  number  of  factors  be  changed, 
the  sign  of  their  product  will  be  changed  (page  67). 

39.  The  product  of  the  sum  and  difference  of  two  quan- 
tities equals  the  square  of  the  first  minus  the  square  of  the 
second  (page  69). 

40.  The  product  of  two  binomials  having  a  common 
term  equals  the  square  of  the  common  term,  and  the  alge- 
braic sum  of  the  unlike  terms  times  the  common  term,  and 
the  algebraic  product  of  the  unlike  terms  (page  70). 

41.  The  product  of  any  two  binomials  equals  the  prod- 


PRINCIPLES.  303 

net  of  the  first  terms,  and  the  algebraic  sum  of  the  prod- 
ucts obtained  by  a  cross-multiplication  of  the  first  and 
second  terms,  and  the  algebraic  product  of  the  second 
terms  (page  71). 

42.  The  difference  of  the  equal  even  poTvers  of  two  quan- 
tities is  divisible  by  both  the  sum  and  the  difference  of  the 
quantities  (page  73). 

43.  The  sum  of  the  equal  odd  powers  of  two  quantities 
is  divisible  by  the  sum  of  the  quantities  (page  74). 

44.  The  difference  of  the  equal  odd  powers  of  two  quan- 
tities is  divisible  by  the  difference  of  the  quantities  (page 
75). 

45.  The  laws  of  the  quotient  in  exact  division  (page  76). 

46.  A  divisor  of  a  quantity  is  one  of  the  two  factors  of 
the  quantity,  and  the  quotient  is  the  other  (page  78). 

47.  A  factor  of  every  term  of  a  quantity  is  a  factor  of 
the  quantity  (page  78). 

48.  The  highest  common  divisor  is  the  product  of  all 
the  common  prime  factors  (page  87). 

49.  The  lowest  common  multiple  of  two  or  more  quan- 
tities equals  the  product  of  all  their  different  prime  factors, 
each  taken  the  greatest  number  of  times  it  occurs  in  any 
one  of  them  (page  90). 

50.  Dividing  one  quantity  and  multiplying  another  by 
the  same  factor  does  not  alter  their  product  (page  93). 

51.  Multiplying  the  dividend  or  dividing  the  divisor 
multiplies  the  quotient  (page  93). 

52.  Dividing  the  dividend  or  multiplying  the  divisor 
divides  the  quotient  (page  94). 

53.  Multiplying  or  dividing  both  dividend  and  divisor 
by  the  same  quantity  does  not  alter  the  quotient  (page  94). 

54.  Multiplying  the  numerator  or  dividing  the  denom- 
inator multiplies  the  value  of  a  fraction  (page  101). 

55.  Dividing  the  numerator  or  multiplying  the  denom- 
inator divides  the  value  of  a  fraction  (page  102). 


304  ELEMENTARY  ALGEBRA. 

56.  Multiplying  both  terms  of  a  fraction  by  the  same 
quantity  does  not  alter  its  value  (page  103). 

67.  Dividing  both  terms  of  a  fraction  by  the  same  quan- 
tity does  not  alter  its  value  (page  103). 

68.  Changing  the  signs  of  both  terms  of  a  fraction  does 
not  alter  its  value  (page  104). 

69.  Changing  the  apparent  sign  and  the  sign  of  either 
term  of  a  fraction  does  not  change  the  value  of  the  fraction 
(page  104). 

60.  Any  common  multiple  of  the  denominators  of  two 
or  more  fractions  is  a  common  denominator  of  the  fractions 
(page  108). 

61.  The  lowest  common  multiple  of  the  denominators 
of  two  or  more  fractions  in  their  lowest  terms  is  the  lowest 
common  denominator  (page  108). 

62.  The  sum  of  two  or  more  similar  fractions  equals  the 
sum  of  their  numerators  divided  by  their  common  denom- 
inator (page  110). 

63.  The  difference  of  two  similar  fractions  equals  the 
difference  of  their  numerators  divided  by  their  common 
denominator  (page  110). 

64.  The  product  of  two  fractions  equals  the  product  of 
their  numerators  divided  by  the  product  of  their  denom- 
inators (page  115). 

65  Canceling  a  factor  common  to  the  numerator  of  one 
fraction  and  the  denominator  of  another  does  not  alter  the 
product  of  the  fractions  (page  115). 

66.  The  quotient  of  two  fractions  equals  the  dividend 
multiplied  by  the  inverse  of  the  divisor  (page  116). 

67.  Canceling  a  factor  common  to  the  numerators  or 
the  denominators  of  two  fractions  does  not  alter  their  quo- 
tient (page  117). 

68.  Eaising  both  terms  of  a  fraction  to  any  power  raises 
the  fraction  to  that  power  (page  120). 

69.  Principles  of  transformation  of  equations  (page  125). 


PRINCIPLES.  305 

70.  Any  term  of  an  equation  may  be  transposed  from 
one  member  to  the  other  if  its  sign  be  changed  (page  125). 

71.  An  equation  with  fractional  terms  may  be  cleared 
of  fractions  by  multiplying  both  members  by  a  common 
multiple  of  the  denominators  of  the  fractions  (page  126). 

72.  The  binomial  theorem  (page  161). 

73.  The  square  of  a  polynomial  equals  the  sum  of  the 
squares  of  its  terms,  and  twice  the  product  of  each  term 
into  all  the  following  terms  (page  163). 

74.  The  cube  of  any  trinomial  equals  the  sum  of  the 
cubes  of  its  terms,  and  three  times  the  square  of  each  term 
into  all  the  other  terms,  and  six  times  tlie  product  of  the 
three  terms  (page  164). 

76.  Dividing  the  exponent  of  any  factor  by  the  index  of 
a  root  takes  that  root  of  the  factor  (page  165). 

76.  Taking  a  root  of  every  factor  of  a  quantity  takes  the 
root  of  the  quantity  (page  165). 

77.  Any  even  root  of  a  positive  quantity  may  be  either 
positive  or  negative  (page  166). 

78.  Any  odd  root  of  a  quantity  has  the  same  sign  as  the 
quantity  (page  166). 

79.  An  even  root  of  a  negative  quantity  is  impossible 
(page  166). 

80.  Extracting  a  root  of  both  terms  of  a  fraction  ex- 
tracts the  root  of  the  fraction  (page  167). 

81.  If  a  number  be  pointed  off  into  terms  of  two  figures 
each,  beginning  at  the  units,  the  unit  of  each  term  will  be 
a  perfect  square  (page  172). 

82.  If  a  number  be  pointed  off  into  terms  of  three  fig- 
ures each,  beginning  at  the  units,  the  unit  of  each  term 
will  be  a  perfect  cube  (page  177). 

83.  Every  pure  quadratic  equation  of  one  unknown 
quantity  may  be  reduced  to  the  form  oi  as?  =  by  in  which 
a  and  b  are  integral  and  a  positive  (page  182). 

84.  Every  pure   quadratic  equation  of  one  unknown 


306  ELEMENTARY  ALGEBRA. 

quantity  has  two  roots,  numerically  equal,  but  opposed  in 
sign  (page  183). 

85.  Every  complete  quadratic  equation  of  one  unknown 
quantity  may  be  reduced  to  the  form  of  aoi? -{-hx  =  c,  in 
which  tty  hy  and  c  are  integral,  and  a  positive  (page  185). 

86.  Every  complete  quadratic  equation  of  one  unknown 
quantity  may  be  reduced  to  the  form  of  x^ -\-px  =  q,  in 
which  p  and  q  may  be  integral  or  fractional,  positive  or 
negative  (page  185). 

87.  The  sum  of  the  two  roots  of  an  equation  of  the  form 
of  x^-\-px:=q  equals  the  coefficient  of  x,  with  the  sign 
changed  (page  194). 

88.  The  product  of  the  two  roots  of  an  equation  of  the 
form  of  x^-\-px  =  q  equals  the  absolute  term  with  the 
sign  changed  (page  194). 

89.  Multiplying  or  dividing  both  terms  of  a  fractional 
exponent  by  the  same  quantity  does  not  change  its  value 
(page  215). 

90.  The  exponent  of  a  factor  in  the  product  equals  the 
sum  of  the  exponents  of  the  same  factor  in  the  multipli- 
cand and  multiplier,  when  the  exponents  are  positive  frac- 
tions (page  215). 

91.  The  exponent  of  a  factor  in  the  quotient  equals  the 
exponent  of  the  same  factor  in  the  dividend,  minus  the 
exponent  of  that  factor  in  the  divisor,  when  the  exponents 
are  positive  fractions  (page  215). 

92.  A  quantity  affected  by  a  negative  exponent  equals 
the  reciprocal  of  the  quantity  affected  by  a  numerically 
equal  positive  exponent  (page  216). 

93.  A  quantity  affected  by  a  positive  exponent  equals 
the  reciprocal  of  the  quantity  affected  by  a  numerically 
equal  negative  exponent  (page  216). 

94.  A  factor  may  be  transferred  from  either  term  of  a 
fraction  to  the  other  if  the  sign  of  its  exponent  be  changed 
(page  217). 


PRINCIPLES,  307 

95.  The  exponent  of  a  factor  in  the  product  equals  the 
sum  of  the  exponents  of  the  same  factor  in  the  multipli- 
cand and  the  multiplier  when  the  exponents  are  negative 
(page  217). 

96.  The  exponents  of  a  factor  in  the  quotient  equals  the 
exponent  of  the  same  factor  in  the  dividend,  minus  the 
exponent  of  that  factor  in  the  divisor,  when  the  exponents 
are  negative  (page  218). 

97.  fl'  X  a'  =  «*"^*'  for  any  positive  or  negative,  integral 
or  fractional,  values  of  a;  and  y  (page  219). 

98.  a'  -^  rt"  =  a'""  for  any  positive  or  negative,  integral 
or  fractional,  values  of  x  and  y  (page  219). 

99.  («')"  =  «*'  for  any  positive  or  negative,  integral  or 
fractional,  values  of  x  and  y  (page  220). 

100.  {a  b)'  and  a'  X  If  are  equivalent  for  any  positive 
or  negative,  integral  or  fractional,  values  of  x  (page  220). 

101.  (x)   and  t;  are  equivalent  for  any  positive  or 

negative,  integral  or  fractional,  values  of  x  (page  220). 

102.  Any  root  of  the  product  of  two  quantities  equals 
the  product  of  the  like  roots  of  those  quantities  (page  223). 

103.  The  product  of  the  equal  roots  of  two  quantities 
equals  the  like  root  of  their  product  (page  223). 

104.  Any  root  of  the  quotient  of  two  quantities  equals 
the  quotient  of  the  like  roots  of  those  quantities  (page  223). 

105.  The  quotient  of  the  equal  roots  of  two  quantities 
equals  the  like  root  of  their  quotient  (page  223). 

106.  No  fractional  radical  is  pure  (page  223). 

107.  Any  quantity  equals  the  wth  root  of  the  nth  power 
of  the  quantity  (page  223). 

108.  The  Wa  and  the  "Va  are  equivalent  (page  223). 

109.  Every  imaginary  quantity  of  the  second  degree 
may  be  reduced  to  the  form  of  ±  a;  a/—  1,  in  which  x 
may  be  rational  or  irrational  (page  234). 


308  ELEMENTARY  ALGEBRA. 

110.  Every  binomial  surd^of  the^  second  degree  may  be 
reduced  to  the  form  of  V a  ±  Vh,  in  which  one  of  the 
terms  may  be  rational  (page  236). 

111.  A  binomial  surd  may  be  a  perfect  square,  and, 
when  it  is  the  square  of  a  binomial  surd  of  the  second 
degree,  one  of  the  terms  is  rational  (page  236). 

112.  A  binomial  surd  with  a  rational  term,  and  the 
coefficient  of  the  irrational  term  reduced  to  ±  2,  is  a  per- 
fect square  when  the  quantity  under  the  radical  sign  is 
composed  of  two  factors  whose  sum  equals  the  rational 
term ;  and  its  square  root  equals  the  sum  or  difference  of 
the  square  roots  of  these  factors  (page  236). 

113.  When  a  binomial  surd  is  a  perfect  square,  the  dif- 
ference of  the  squares  of  its  terms  is  a  perfect  square,  and 
is  equal  to  the  square  of  the  difference  of  the  two  factors 
described  in  Prin.  112  (page  237). 

114.  Principles  of  transformation  of  inequalities  (page 
249). 

115.  The  sum  of  the  squares  of  two  unequal  quantities 
is  greater  than  twice  their  product  (page  249). 

116.  The  sum  of  the  squares  of  three  unequal  quantities 
is  greater  than  the  sum  of  their  products  taken  two  and 
two  (page  249). 

117.  The  ratio  equals  the  antecedent  divided  by  the 
consequent  (page  252). 

118.  The  antecedent  equals  the  ratio  times  the  conse- 
quent (page  252). 

119.  The  consequent  equals  the  antecedent  divided  by 
the  ratio  (page  252). 

120.  Multiplying  the  antecedent  or  dividing  the  con- 
sequent multiplies  the  ratio  (page  252). 

121.  Dividing  the  antecedent  or  multiplying  the  con- 
sequent divides  the  ratio  (page  252). 

122.  Multiplying  or  dividing  both  terms  of  a  ratio  by 
the  same  quantity  does  not  alter  its  value  (page  252). 


PRINCIPLES.  309 

123.  An  infinitesimal  divided  by  a  finite  constant  is  an 
infinitesimal  (page  2G8). 

124.  An  infinitesimal  multiplied  by  a  finite  constant  is 
an  infinitesimal  (page  2G9). 

126.  An  infinitesimal  divided  by  an  infinitesimal  may 
be  any  finite  constant  (page  269). 

126.  An  infinite  divided  by  a  finite  constant  is  an  in- 
finite (page  269). 

127.  An  infinite  multiplied  by  a  finite  constant  is  an 
infinite  (page  269). 

128.  An  infinite  divided  by  an  infinite  may  be  any  finite 
constant  (page  269). 

129.  A  finite  constant  divided  by  an  infinitesimal  is  an 
infinite  (page  269). 

130.  The  product  of  an  infinitesimal  and  an  infinite 
may  be  any  finite  constant  (page  269). 

131.  A  finite  constant  divided  by  an  infinite  is  an  infini- 
tesimal (page  270). 

132.  In  an  arithmetical  progression, 

l  =  a-^{n-l)d  (page  273). 

133.  In  an  arithmetical  progression, 

>S'=(^  +  «)|  (page  274). 

134.  In  a  geometrical  progression, 

lz=iar*-^  (page  280). 

135.  In  a  geometrical  progression, 

136.  In  an  infinite  series,  S  =  z (page  286). 


APPEIfDIX. 


Highest  Common  Divisor  by  Successive  Division. 

Definitions  and  Principles. 

1.  If  one  quantity  be  divided  by  another,  then  the 
divisor  by  the  remainder,  then  the  next  divisor  by  the 
next  remainder,  and  so  on,  until  the  division  terminates, 
the  process  is  called  Successive  Division, 

2.  Since  a  is  a  divisor  ot  ax  and  also  of  nax,  it  fol- 
lows that, 

Trin,  1, — Any  divisor  of  a  quantity  is  also  a  divisor 
of  any  number  of  times  the  quantity, 

3.  Since  «  is  a  common  divisor  of  at  and  aCy  and  also 
a  divisor  of  ai  ±  ac,  it  follows  that, 

JPrin,  2. — A  common  divisor  of  two  quantities  is  also  a 
divisor  of  their  sum  and  of  their  difference, 

4.  Theorem, — The  last  divisor  obtained  by  the  succes- 
sive division  of  two  quantities  is  their  highest  common 
divisor. 

Demonstration :    Let  A  and   B  A)  B  (q 

represent  any  two  quantities.     Di-  ^  ^ 

vide  B  by  J.,  and  let  the  quotient  — —.         .    , 

be  q  and  the  remainder  R ;  divide  ^  )  ^    \Q 

A  by  E,  and  let  the  quotient  be  q'  ^^  Q 

and  the  remainder  B' ;  divide  i^  by  R' )  R  {  q" 

R',  and  let  the  quotient  be  q"  and  R'  a" 

the  remainder  zero.     Prove  R'  the  — pp" 

H.  C.  D.  of  A  and  B.  " 


APPENDIX.  311 

1.  R  is  a  divisor  of  R,  since  the  division  has  terminated ;  hence  it 
is  also  a  divisor  of  R  q'  [P.  1]  and  of  R'  +  R  q',  or  A  [P.  2] ;  and, 
therefore,  ot  Aq  [P.  1],  and  of  i2  +  ^  ^,  or  ^  [P.  2] ;  therefore,  R' 
is  a  common  divisor  of  A  and  B. 

2.  There  can  be  no  higher  common  divisor  of  A  and  B  than  R' ; 
for  if  there  could  be  it  would  be  a  divisor  of  A  and  B,  and  hence,  too, 
ot  Aq  [P.  1],  and  ot  B  —  Aq,  or  R  [P.  2],  and,  therefore,  of  R q' 
[P.  1],  and  ot  A  —  Rq\  or  R  [P.  2] ;  that  is,  a  quantity  of  a  higher 
degree  than  R  would  be  a  divisor  of  R',  which  is  impossible.  There- 
fore, R  is  the  H.  C.  D.  of  A  and  B. 

6.  Since  the  highest  common  divisor  equals  the  product 
of  all  the  common  factors,  it  follows  that, 

PHn.  3. — Either  of  two  quantities  may  he  multiplied 
or  divided  by  a  factor  not  found  in  the  other  without 
changing  their  highest  common  divisor. 

Problem.     To  find  the  highest  cominoii  divisor  by 
successive  division. 

niustratioiis.— 1.  Find  the  H.  C.  D.  of  : 

%3^-^a?-14.x-{-Z  and  ^x'' -14:3?  -  ^x-^% 

Foniit 

2a:* --9a;3- 14a; +  3)6ar*- 28  a;3_  13^^4(3 
6^*j-2?V--42^-l-9 
-l)-x^-\-%^x-6 

a;»-24a;  +  5)2ic*-    ^a? -l^x-\-Z{'Zx-^ 

-  9a?-\-AS3?-    24a;+   3 

-  9a? +216 a; -45 

48)48  3:^-240  a:  +  48 

a?-      5x-^    1 
H.  0.  D. 

Q^-bx-\-l)a?-24:X  +5(2;-h5 

3?  —     53?-\-X 

5ic2_25a;  +  5 
6a?-'25x-\-6 


312  ELEMENTARY  ALGEBRA. 

Solution :  Taking  the  second  polynomial  for  the  dividend,  we  ob- 
serve that  the  first  term  of  it  is  not  divisible  by  the  first  term  of  the 
divisor;  we  therefore  multiply  the  dividend  by  2  [P.  3],  and  then 
divide  and  obtain  for  the  first  remainder  —  a;^  +  24  a;  —  5.  We  now 
divide  the  remainder  by  —  1  [P.  3],  and  divide  the  divisor  by  the 
result,  obtaining  for  the  next  remainder  48  x^  —  240  x  +  48.  We  reject 
the  factor  48  from  this  remainder  [P.  3J  and  divide  the  previous 
divisor  by  the  result,  and  obtain  no  remainder.  The  last  divisor, 
x^  —  ^x  +  \,  is  the  H.  C.  D.  (theorem). 

6.  If  one  polynomial  can  be  factored,  its  factors  may  be 
made  ayailable  in  factoring  the  other  by  trial.  The  first 
term  of  a  factor  is  always  a  divisor  of  the  first  term  of  the 
polynomial,  and  the  last  term  of  a  factor  a  divisor  of  the 
last  term  of  the  polynomial. 

2.  Find  the  H.  0.  D.  of  : 

^4_3^_28  and  x'>-'2:^-\-1  x^-lO^-^l'^x-^ 

Solution:  The  factors  of  x^  —  ^x^  —  2S  are  x^  —  l  and  a;2  +  4; 
x^  —  1  is  not  a  factor  of  x^  — 'Z  a^  +  11  oi^  —  10  x^  ■\- V^  x  —  %,  since  7  is 
not  a  divisor  of  8 ;  if  the  two  polynomials  have  a  common  divisor,  it 
must  therefore  be  x'^  +  4.  By  trial  we  find  that  a;^  +  4  is  a  divisor 
of  the  second,  and  is  therefore  the  H.  C.  D.  of  the  two. 

7.  Since  each  remainder  is  a  number  of  times  the 
H.  0.  D.,  it  is  sometimes  more  convenient  to  use  the  re- 
mainders, or  a  remainder  and  one  of  the  quantities,  than 
to  use  the  polynomials  themselves  in  the  progress  of  the 
work. 

3.  Find  the  H.  C.  D.  of  :^-Qx^-x-\-m  and 

Form. 

^_6a;2-a:  +  30):z;3  +  9a:3  +  26iz;  +  24(l 
a^-Qx^-      g;  +  30 
3)152;^  +  27a;-    6 


^7?-^    ^x-    2  = 
{5x-l){x-\-  2) 
H.  C.  D.  =  a:  +  2 


APPENDIX.  313 

EXERCISE     1. 

Find  the  H.  C.  D.  of  : 

1.  a:^  +  3 7?^ 3a;  +  2anda:3_^2_^__2 

2.  7?  — b'jr-\-Zx-\-4i  and  7:^-{-bx^—'7x  —  Q 

3.  2rr*-a;3  +  2a;2_j_^_l  and  %a^ -^7? -\-bx -% 

lbo^-^OQ^-4.7^-\-Qx-^ 

5.  2  a;*  -  3  a:^ +  4^:2  +  3  a; -6  and 

2a;*-3a:3_|_8a:2_3^_|_6 

6.  6a*  +  7a'4-7a2  +  3a  +  l  and 

14a*  +  a3-j_8a2_^_^2 

7.  a:24-8a;  +  15  and  a:^  -  3a;2- 10a;  + 24 

8.  6a;-  +  5a;-6  and  8  a:^  -  22  a;^  __  21  a;  +  45 

9.  7?-^^x'-\-^x-^l  and3a:3-a;2_ii^_.j' 

10.  7^-Qx^-\-Qx'-Zx-10  and 

3  a:*  -  13  a;3  -  11  a;2  4-  8  a;  -  15 

11.  42^3  +  8y2  +  8?^4-4  and  7«/3-14/  +  21 

12.  a^2  +  ^>»2  and  a^^  _^  «» ^s  +  5^6 

13.  a^-f  2a^  +  J2_^2  ^ud 

a2-«J_2^>2_^4«c  +  Jc  +  3c2 

14.  a:*  +  a:«/  +  y*and  xf -\-^7?y -^-^^x"" y"" -\-dxy^ -\-f 

15.  3«^  +  201fl2_|_i98and  Sa^  +  lOa^  +  lOa^-j-iOa  +  S 

16.  Qax^  —  ^axy-l%ay^2iTidiQhx^  —  lQhxy-\-^hy^ 

8.  T^e  highest  common  divisor  of  three  quantities  may 

he  obtained  hy  finding  the  highest  common  divisor  of  two 

of  them,  and  then  the  highest  common  divisor  of  that  and 

the  third  quantity. 

Demonstration. — Let  x,  y,  and  z  be  three  quantities,  and  m  the 
H.  C.  I),  of  X  and  y,  and  n  the  H.  C.  D.  of  m  and  z ;  then  will  m  be 
the  product  of  the  factors  common  to  x  and  y,  and  n  the  product  of 
the  factors  common  to  m  and  z,  or  n  will  be  the  product  of  the 
factors  common  to  x,  y,  and  ^,  which  is  their  H.  C.  D. 


314  'ELEMENTARY  ALGEBRA. 

EXERCISE    2. 

Find  the  H.  C.  D.  of  : 

1.  x^  —  xy^,  ^y-\-y^i  and  x^y  —  xy^ 

2.  x^  -\-xy  -\-y^,  ^ -\-x^y^  -\- y^,  and  x^ -\- xf^  y^ -^  y^ 

3.  2  a:2  +  3  a;  +  1,   2  2;2  +  5  2:  +  2,   and 

2ic3_j_5a;2_4^_3 

4.  3ic3-17a;  +  10,  ^t^ -^x"" -Zx^^,  and 

3:z:*-2a:3  4-3a;-2 

5.  2a:3  +  7a:2  +  8aJ  +  3,   2ar^  -  a;^  -  4rz;  + 3,   and 

2:c5  +  3ic*  +  2a;3  +3a;2  +  2a;  +  3 

6.  4a;3  +  4a;2-36rr-36,  4rc^  -  4a;2  -  36a;  + 36, 

and  2a;3  +  6a:2  — 2a;— 6 

7.  ic*  +  a;3  — 8ar  — 9a;  — 9, 

a;5  +  3a;*  +  a:-^  +  3a;2_^^^3^  and 

a:5  +  22;*  +  a.'3  +  2a;2  +  a:  +  2 

8.  12a;3-2x2-3a;  +  2,   ISa;^  -  9a;2  -  8a;  +  4,  and 

36  ic*  -  25  a;2  +  4 


Lowest  Common  Multiple  of  Quantities  not 
readily  factored. 

9.  To  find  the  lowest  common  multiple  of  quantities 
not  readily  factored. 

Theorem, — The  lowest  common  multiple  of  two  quan- 
tities equals  their  product  divided  hy  their  highest  common 
divisor. 

Demonstration.— Let  c  equal  the  H.  C.  D.  of  A  and  B. 

Suppose  —  =  a:,  and  —  =  y ; 

then         A  =  c  X  X,  and  B  =  c  y.  y. 

Now  X  and  y  are  prime  to  each  other,  since  c  is  the  product  of 

all  the  common  factors. 

T    n  Tit                             A               .      B      A  y.  B 
.'.    Li.  ij.  M.  =:cxxxy  =  Axy  =  Ax— = 


APPENDIX.  315 

lUnstration.— Find  the  L.  C.  M.  of  6  ar^  -f  13  a;  +  6 

and  6a:3_|.9a^_|_3^_j_12. 

Solution  :  We  find  the  H.  C.  D.  to  be  2  a;  +  3.    Therefore,  the 

3a;»  +  4 
l^  ^  ^l  _(ga;*^1-4^^3^a^(6a:*  +  9a;*  +  8a;  +  12)_ 
'  ^  '  ~  2^ir-K3 

(3rK«  +  4)(6a:»  +  Oa;'  +  8a;  +  12). 

EXERCISE    3. 

Find  the  L.  C.  M.  of  : 

1.  6a:2_^i4a._j_8  a^^  8a:2_|_6^_20 

2.  a;3  +  6ar^4-6a:  +  5  and  ^7? -\-^x^ -^^x-^1 

3.  o^  -  2  a  -  1  and  fl3  +  2  «2  4-  2  «  +  1 

4.  a*H-2a2-^.9  and  1  a^ -11  aJ" -\-l^ a -^^ 

5.  2  a^  —  a:*  y  +  ^  y^  +  3/^  a^d  2  o^  -{-^  x^  y  -{-^  x  if  -\-  tf' 

6.  a:^  +  2a:^y4-2a;^^  +  y^  and 

a;3  +  3:r2|^  +  3.r/  +  2?/=* 

7.  3a;3  4-5a:2  +  3a;  +  l  and  3a:*  +  2a:3  +  4a;2  4- 2a;+ 1 

8.  2a:3_^^a;2_j_^2^_^3  and  3a;3  +  4aa:2  _^  4^2^^^3 

9.  Za?  —  l^a?-\-14.x-Q  and  6a;3  +  a;2  _  g.^^  6 

10.  4««-4a*-29a2_21  and  4 a'^  +  24 a*  +  41  «2 _|_ 2I 

11.  20;z*  +  ;z2_i  ^nd  25;2*  +  5^^  -  ;2  -  1 

12.  3a:*  +  5a;3^52.2_^5a._^2  and 

3  re*  -  a;^  +  a:^  __  ^  _  2 

13.  6a:*  +  17a:3-10a;  +  8  and 

9a;*  +  18a:3_^2_9^^4 

14.  2a-^-4a2_  13^-7  and  Ga^  -  11  a^  _  37a  -  20 

15.  6m3+15m2-67?i4-9  and  9^^+ Gm^- 51  w  +  36 

16.  ?i*  —  an^  —  a^n^  —  a^n  —  '^a*'  and 

37i3-7a?i2-|-3a2  7i-2a3 


ANSWEES 


Exercise  1. 


1.  9  units,  9  tens,  9  fives;  9  times  the  number. 

2.  9  times  a,  13  times  a  3.  15  a,  19  a  4.  9  J,  155 

5.  27,  36         6.  10  m,  20,  50         7.  13^,  39,  78        8.  10^,  10  n 
9.  5  tens,  5  twenties;  5  times  tlie  number;  5a,  5m 

10.  8x,  7y         li.  5a,  15,  35  12.  4a,  20,  32  13.  12,  35" 

14.  48,  90  15.  mn,  pq,  xyz  16.  pqr,  24,  60 
17.  210,  108                    18.  2,  3,  6  19.  3,  4 
20.  3,  7                            21.  4,  3                         22.  8,  14 

23.  2.  5;  3,  5;  3,  7;  3,  a;   5,  a;;  a,  y;  x,  z 

24.  2,  2,  3;  2,  3,  3;  2,  3,  5;  2,  5,  a;  5,  a,  6;  x,  y,  z 

25.  2,  7;  3,  7;  5,  m;  c,  d  26.  2,  5,  x\  5,  a,  y;  p,  q,  r 
27.  4,  8,  16         28.  9,  64,  16         29.  16,  27  30.  8,  27,  64 
31.  36,  16            32.  a,  a;   x,  x,  x;   m,  m,  m,  m;  x.  x,  y,  y,  y 
33.  2,  3,  4,  a,  x             34.  2,  a-,  3a              35.  4,  5,  a,  ax 
36.  3,  4,  a,  ax               37.  4,  7                     38.  3,  4 

39.  x,  a,  c  40.  6,  4  41.  w^,  n^,  mn,-,^ 

Exercise  2. 

1.  8,  17  2.  rc  +  y,  a:  +  y  +  2,  2a  +  3^  4j:  +  5//  +  62 

3.  5,  6  4.  m-n,  2a-3  6,  5x2-72/*,  a:^-?/' 

6.  9,  3,  24,  27,  9,  189,  54,  21  7.  20,  8,  12,  0,  18,  22,  18,  46,  40 
8.  2a6«,  4a6«,  7o6«;    3a'b,  GaH,  SaH,  9a«5;    5aH\  Qan\ 

9a«6«  11.  32  12.  a  +  b  13.  a-b  14.  ab 

15.  J        16.  (a  +  b)^         17.  a2  +  5-'  18.  (a  +  Jf         19.  a»  +  b^ 

20.  (a  +  6)(a-6)  21.  ab{a-b)  22.  ^ 

15 


318  ELEMENTARY  ALGEBRA. 


Exercise  3. 

2.  5  a  ct.                       3.  $3« 

4.  (3  a  +  h)  ct. 

5.  10  re  mi.                    6.  (Sa  qt. 

7.  (10-c)  ct. 

8.  %{y-m)                  9.  %{^x  +  4:y) 

10.  %Zm 

11.3a    8.                  12.?^^^^ 

X 

13.  f  |, 

14.  ^                   15.  1»«                 16.  1 

6                           M                         6 

17.  %m{r-n) 

18.  I*'"^            19.  "f               20.  $a 
a                        0 

(^-)      -^-  ^ 

22.  (2  a +  20)  yr.,   (2a-10)yr. 

23.  {m—7i)c  or  {n—m)c 

^-  *ll           ^^-  w 

^^■<^-m 

^'•*(— 1^)''     ^'-''^ 

-2 ex         29.  ^ — ct. 

a  +  c 

3°-      /        .xJ"--                     31.     j3 

32.^ 

a 

^^•C-i'')i»".(^)'«'> 

'^^e^'- 

3^-(^-^)««'(^-l)««^-' 

'^■^"^ 

37.  (2;.v-s2)  sq.  rd.                                  3£ 

1.  —  tons. 

39.  V^'^  ft.                                           4C 

^    172Spqr 
x^ 

43.$(.+  ^gi„-a 

41.  V«'+^'^  yti.           42.  Vc'-tt' 

Exercise  4. 

14.   +4  lb.                    16.  -15  bii. 

17.   +(a— 5),    —{b—a) 

18.   +{:x  +  y)                  19.  -{a  +  h) 

20.   +  a  &  sq.  rd. 

Exercise  5. 

2.   +61  lb.            4.  -45  cows            6. 

-30  lb.              7.  +8 a 

9.   +5                  10.  0                          11. 

+  4:X                 12.  — 4wi 

Exercise  6. 

1.    +20 a                2.  -25a;                3.  - 

-9.ry                4.  —4a  6 

5.   +2aa:               6.  -(Sx^y              7.  - 

-3wn               8.  —Spq 

ANSWERS.  319 

9.  2{a  +  b)                         10.  2(:m-?i)  11.  -{x^  +  i/) 

12.  -8(.r  +  v/)2                     13.  Uan^  14.  -4a: 2/2 

15.  —dia-m)                    16.  8«  17.  0 

18.  -7a 6                           19.  -(Sm^n^  20.  -4{a  +  b) 

Exercise  7. 

1.  ;i-i-i/  +  2                      2.  2x—dy  +  4:Z  3.  5a— 36— 2c 

4.  7r+6»— 5m             5.  4h  +  cd—ab  8.  4xy—3ab 

6.  7ab—4cd+5ac—Gbd  +  4:am  9.  4a  +  86 

7.  7y0— a:y— 4a;2  +  9m^— 6wa:  10.  a;  +  7/  +  142; 

11.  6/?i2  +  „i;i_3;i2  12.  0 

Exercise  8. 

1.  (a  +  b  +  c)x  2.  {a—d  +  m)yz 

3.  (2a  +  3  6-4c)?/  4.  (2-a-6)a:2; 

5.  (2a+3  6  +  4c— rf)a:y  6.  {n—m+p—q  +  r)ab 

7.  (— a  +  26— 3c)a:y  8.  (3a6— 4&c  +  6cc?— 5a<f)my 

9.  {a  +  b){c  +  d)  10.  (a  +  &-c)(a:  +  y  +  2) 

Exercise  10. 

1.  +4a            2.  -3a                  3.  -4a  4.  5a;          5.  136 

6.  +146          7.  -16a6             8.  -ISxy  9.  5a  +  126 
10.  a  +  6           11.  ^mn—xy       12.  — a^a;'^  13.  — m^n' 

14.  13a;2y2                     15,  Sxy-7mn  16.  Gmx-m^n 

17.  c«  +  tZ2                     18.  m«-7i2  19.  -(a+a;) 

20.  9(0:2-/)                 21.  lG(a:-2/)«  22.  (a  +  6)y« 

23.  {d-c)x^                 24.  (w-w»)a;  25.  (2a-3  6)a; 

Exercise   13. 

1.  +Ga2                2.  -8j;2                3.   +IO3/2  4.  -ISm^ 

5.  +12a:*              6.  -12a'              7.  6a6  8.  ISa:*^' 

9.  —lOx^y^z^                10.  — 21a3  6'  11.  28 a m^n* 

12.  -18a:«y«2<                13.  (a  +  6)'  14.  (m-rt)' 

15.  12(a-6)»                    16.  a'(a  +  6)*  17.  -24mp^q^r 

18.  -140r»s«2»               19.  14a«6«  20.  6  a"  6' 

21.  -X*                            22.  12  m«n»  23.  84  a:^  3/' 

24.  (a  +  6)»                        25.  (a-3  6)'»  26.  3a(a:-y)» 


320  ELEMENTARY  ALGEBRA. 

Exercise  15. 

L  10a2_l5«Z,  +  20ac  2.  -^a^h  +  Qa:^h'^-Zah^ 

3.  25a:4y3+15a;3  2/4-10a:2  2/5        4.  14a3a;3  2/_i2a2a;2  2/'^  +  18a2a;2/3 

5.  —  2. -r^  2/2  +  2  a;4  2/3^2. ?;3y4 

6.  15  a3  ^,2  a:4  y  +  25  a^  J3  ^4  y3_35  aV'x^y^ 

7.  30m8  +  25m'-20m«  +  15w5— 25m4 

8.  aS&3_a7&4^^6^,5_a5^,6  +  ^4J7_a3J8 

9.  -30 a2 :;:5 ^2  +  25  a^ a^  ?/3-35  a^ a:V  +  25  a^ a;2 2/3-25  a'^xy^ 

10.  a^bc{x  +  y)  +  ab^c{m  +  7i)—ab  c^  (p  +  q) 

11.  2(a  +  5)a.-42/2-2(a-6)a;3  2/3  +  2a&a;V 

12.  S  a{x  +  yy-4:b {x  +  yf-2  c (x+yf 

13.  4a:(a  +  &)3-62/(a  +  5)^  +  82(a  +  Z')5 

14.  aHix-yf-a'^b^x-yf  +  ab'^ix-yy 

15.  p'^qx^{p  +  qf—p  q^  x^  {p  +  qf  +p'^  q^x{p  +  qy 


Exercise 

19. 

1.2  a 

2.  -3a& 

3.  -Sa'b 

4.  dx'^y 

5.  dm*y 

6.  -In^b^xy^ 

7.  82/3 

8.  3i«2  2/2 

9.  (a +  6)3 

10.  -{m-nf 

11.  Sa(a+xy 

12.  3  2/ (a- 6)* 

13.  52(a;  +  2/)* 

14.  -4a;2(a;2- 

.2/2)3 

15.  -6m3(a3  +  63)3 

16.  3a:2  2/2  22 

17.    -6./:2  2/26 

18.  -WaHH^ 

19.  -2xyz 

20.  -9a5^,6ca 

'^ 

21.  0 

Exercise  21. 

1.  a^  +  a  2.  ;/:3  +  2a:  3.  2a-Sb  4.  2xy  +  xHf 

5.  2a-3&      6.  a3  +  <^2_^_i      7.  2a:2  +  3a;  +  4     8.  2a-3a6  +  & 
9.  -2a  +  36-5         10.  262_3a  6  +  7a2         11.  a:2-2a;2/-32/2 
12.  —a^oi^  +  a'^x'^—ax+1  13.  2m27i3_3,,^i,^;2  +  4^i_5y;i 

14.  2a'^{a  +  b)  +  'da{m  +  n)—4:{p  +  q) 

15.  (a  +  3:)2— (a+a:)  +  l  16.  2a-3  6  +  4c  17.  —x-y  +  z 
18.  a:2/-a:2  +  2/^                                19.  {a  +  bf-x\a  +  b)-\-y^ 


Exercise  25. 

1.  a;=8  2.  a:=3  3.  a:=6  4.  .t=3 

5.  cc=i  6.  x=-2  7.  a;=2|-  8.  2:=5i 


ANSWERS.                                       321 

9.  3:= 12               10.  : 

r=24              11.  a:=240             12.  x=^~ 

do 

13.  x=~               14. 

x=\\             15.  x=18              16.  a:=36 
4 

Exercise  26. 

2.  80,  20                  3.  144,   180,  30                  4.  $20,   $30 

5.  $1000,   $1800,   $2400                   6.  10^  32 1,  64^ 

0              0              0 

7.  IGO,  240,  360 

8.  $500,   $2500              9.  48 

10.  $30,   $180 

11.  IC  yr.,  24  yr.          12.  7,  56 

13    ^^     ^^ 
"•  23'  299 

14.  144                           15.  22:^ 

16.  48  yr. 

17.  21  A.,  27  A.           18c  $225,   $275 

19.  20  yr.,  55  yr. 

20.  40,   GO 

21.  $450,   $1050,   $2000                  22.  $1725,  $1475,   $1G00 

23.  $150 

24.  $90                           25.  $415^ 

26.  $1000 

27.  GO                              28.  16 

29.  40 

30.  10  yr.                       31.  30  yr. 

32.  GO,   100 

33.  120 1  A.                  34.  10  yr.,   15  yr. 

35.  $3G00,   $4800 

36.  360,   405               37.  2400  bu.,  2500  bu. 

38.  $240 

39.  50  ct.                        40.  $28 

41.  8,   20 

42.  $40                       43.  78|,  97i|,  93|i 

Exercise  27. 

1.  6a  +  46 

2.  7a  +  lU-9c 

3.  21x—7y  +  5z 

4.  nx^-7if-dxy 

5.  cd—ad  +  e 

6.  8.ry-2  2« 

7.  7  m'^-rs  Tw  7t  +  23  /i«                      8.  3  a  x'^-S  b^  if 

9.  3(A-  +  2/)  +  (w+n) 

10.  \r)ai2)  +  q)  +  3b(p—q) 

Exercise  28. 

L  _a;«+9y« 

2.  4a:'+a:y— Oy 

3.  _3a  +  136-16c 

4.  6m»n«-15w7i-3n« 

6.  hj*-\2x^y  +  ^y* 

6.  y*-12y^+13y-7 

7.  a;«-a:+15 

8.  nx^-7xy-10y» 

9.  2w»-7w«-16n' 

»                       10.  -4.r'+7^*— z+15 

1. 

9x 

6. 

dx 

11. 

5xy 

15. 

5a  +  l5x 

322  ELEMENTARY  ALGEBRA. 

11.  (.r  +  ?/)+14(x— 2/),  or  15a;— 13y  12.  —x  +  ^y—4.z 

13.  4^2_2^2/  +  52/2  14.  9rt3  +  2«2  +  6a  +  5 

Exercise  29. 

2.  2a  3.  la—h         4.  -2a;  +  5y  5.  —2m 

7    — 3a  +  5        8.  a:y  9.  x  +  by  10.  2a;  +  5y 

12.  4  13.  -6  14.  2x-2y-z 

16.  1  17.  5  a: 

Exercise  30. 

1.  {2a  +  db)  +  {5c-2d)  2.  (a-2&)  +  (c-2<Z) 

3.  (m  +  w)-(^— ^)  4.  (3m-2w)-(4^— 3^) 
5.  («-&)  + (c-(Z)-(e-/)             6.  (2a-3&)-(4c-2c?)-(5e-6/) 

7.  (a;— 2/)  +  (22— 3v)— (6w— 4z<;) 

8.  (5j9-3^)  +  (52-4m)  +  (2n-6r) 

9.  {a-b  +  c)-{d+e-f),  (2a-db-4c)  +  (2d-5e  +  Gf), 
{x—y  +  2z)—{Sv  +  Qu—4w),        {5p—Sq  +  5z)—{4:m—2n  +  6r) 

10.  (2m-3n+4:a)—{Qb-7c  +  2d)-{4:e-g  +  2h) 

11.  (4a-2&-3c)-(4c;-5e-6/)  +  (7^-2A  +  4/) 

12.  (2i?-3g  +  4r)-(2s-5/-G«)-(7v-2M>  +  6y) 

13.  1.  {2m-{Sn-4:a)}  —  \(jb-{7c-2d)}  —  {4:e-(g-2h)} 

2.  J4a-(26  +  3c)i  — |4c^-(5e  +  6/)i  +  ]7^-(2A-4?)} 

3.  \2p-{Sq-ir)\  —  \2s-{5t  +  6u)\  —  \7v-{2w-Gij)\ 

14.  ■|.T_(2/_£)}  — |m-(?i-^)}  +  ]g-(r  +  s)} 

Exercise  31. 

1.  a2_j2  2.  a2  +  2a6  +  &2  3.  a^-2ab  +  b^ 

4.  4a2_9^,2  5.  a;2 2/2 +  6 a; 2/- 72  6.  a;2-aa:— 6a:  +  a6 
7.  &C  +  61C— ca;— a;^                             8.  m^  +  m^n^  +  m^n^  +  n* 
9.  12a*-10an-12b^                     10.  63a4_23a2  J2_56^,4 

n.  abcy—acdx—b^dy+ bd^x  12.  a:6  +  l  13.  a^-b^ 

14.  .r«— 8  15.  a'2^1  16.  729 +  c«  17.  b^  +  Mc^ 

18.  6m4  +  4m3n-9m'^w2— 6m7i3  19.  36a.-4y4_49^24 

20.  27a3_i728  21.  a^—y^  22.  a^-^s 

23.  9x^  +  dx''y^  +  4:y''  24.  25  a'^-ga^js^  16^,4 

25.  81a:4-16;c2  2/'-^  +  8a-y='-2/^  26.  4  a;^  +  56  a.-^  +  324 

27.  a6_32&5  28.  243x^  +  32  jf  29.  a;8  +  a^i/'+2/*' 

30.  6a;''  +  13a.6y  +  6a;\'?/'  +  17a-4.y3_i3^;3,/4^.n^2y5_9^^4  +  9^7 


ANSWERS.  323 

Exercise  82. 

1.  a  +  6            2.  x  +  4:            3.  x-5  4.  .t^  +  G            5.  .r  +  y 

6.  x^  +  xij  +  if                 7.  a;*-3a:y  +  9?/2  8.  a-'-a^i^  +  i* 
9.  a^  +  an  +  ab^  +  h^              10.  4a:«-6.xi/  +  9?/  11.  2x-Q 

12.  2a:-3a                    13.  2a:-5a  14.  ma^+20x^y^  +  25y* 

15.  16;i-4-8.x»7/  +  4x2/-2a:y3  +  2/4  15.  4:7n^  +  ()m^n^  +  dn'^ 
17.  a;«-a:2/  +  !/^               18.  3:^  +  2  a; +  4  19.  a'^-aH^  +  b^ 
20.  4w27t44.0m3n3  +  9m4w2  21.  x^+x+1 

22.  3a.-»  +  2^2  +  l            23.  a;«-.r-3  24.  4:a*  +  Qa^-2 
25.  2a:— 2/  +  2                  26.  x—2y—Sz  27.  2a:— 3?/— 2 
28.  2a:  +  3//  +  2               29.  x^—xy—if  30.  a  +  &— c— <i 

Exercise  34. 

1.  a:=3,  y=5                2.  a-=5,  y=:l  3.  ;i=:5,  2/=3 

4.  a:=-2,   y=2             6.  a;=9,   y=8  6.  a:=-3,   i/=-4 

7.  a:=10,   y=0               8.  a:=5,   y=3  9.  a:=0,   ^^=0 
10.  a:=2^,  3^=3^        11.  a-=-5,    y=^  12.  a;=7,  y=9 

13.  a:=12,  2/=24 

Exercise  35. 

1.  $400,   $100               2.  15  yr.,  20  yr.  3.  2  ct,  1  ct. 

4.  56  lb.,  30  lb.            5.  $2000,  $1000  6.  $42,  $26 

7.  18,  20                        8.  25  ct.,  50  ct.  9.  60  ct.,  40  ct. 

10.  $2,   $1                      11.  $6,   $40  12.  80  ct,  90  ct. 

13.  $2,   $1                      14.  14  ct.,   24  ct.  15.  40  A.,   200  A. 

16.  GO  A.,   120  A.         17.  30,  20  18.  $11G,   $166 

Exercise  36. 

1.  a'o^/'c'o           2.  IQa^b^c^          3.  — 27aH9c«  4.  Ux^y*z* 

5.  81a:«y»2^                   6.  -Q4:m^n^x^  7.  32a'^6Wc6c;» 

8.  81a:"  y"  2^                  9.  32x2'>2/'*2»<>  \0.  Q2^  mf^ 'n}^  z^ 

11.  {a  +  hfic  +  df          12.  m4.A:«(a  +  6)»«  13.  21{a-\-bf{x-yf 

14.  d^^b^c*{m-{-nf        15.  8 a« &» (x*  +  yY        16.  m^{x-yy^{x+yf 

17.  108a'9  6»4ci6             18.  6y*2  19.  IQ'^j^y^^z^ 
20.  13a:4y4  2.6                 21.  0  22.  d2x^y^z-* 

23.  -250a:"y"2»8         24.  -48  a:"  7/1*2" 


324  ELEMENTARY  ALGEBRA. 

Exercise  37. 

1=  x'^  +  2xy  +  y'^  2.  x'^—'^xy  +  y^  3.  m'^  +  2mn  +  n^ 

4.  m^—2mn  +  n^  5.  a;2  +  8a;+16  6.  x^—Ux  +  i9 
7.  x^  +  2ax  +  a^              8.  x'^—2ax  +  a'^              9.  4:X^  +  4:Xy+y^ 

10.  9x^-Qxy  +  y^        ll.  4:a^  +  12ab  +  9b^     12.  25a^-d0ab  +  9b^^ 

13.  4tx^  +  S2x  +  G4:        14.  9a;2-30a;  +  25  15.  25  +  20x  +  4:X^ 
16.  dQ-d6x  +  9x^        17.  x'^y^  +  2xy  +  l         18.  a;2y2  +  i0xi/  +  25 

19.  l-2c<?  +  c2(;2  20.  l  +  2a;2/  +  a:2y 

21.  an-^-2abcd  +  c^d^  22.  a;2y  +  2a;.v22  +  2/2  22 

23.  4j92^2_i2j9^r+9r2  24.  a*  ^,2  ^  2  a^  &3 ^.  ^-i  ^,4 

25.  4a^2/2_i2a;3  2/3  +  9a;V  26.  («  + J)2_2(a  +  5)  +  l 

27.  ia-bf-2(a-b)  +  l  28.  (a;H-2/)2  +  2(;c  +  ?/)2  +  22 

29.  (a;+2/)2-2(a;+2/)2  +  22  30.  a^-2a{b  +  c)  +  {b  +  c)^ 

Exercise  38. 

1.  a^  +  Sx^y  +  dx7f  +  7f  2.  x^-Sx^y  +  dxy^-y^ 

3.  2/3  +  62/2  +  122/  +  8  4.  2/3_62/2^.i22/-8 

5.  l  +  3a;+3a;2  +  a;3  6.  x^-Sx^  +  Sx-1 
7.  a;32/^  +  3.i2^2^3^y_^l  8.  8  +  I22  +  622  +  23 

9.  a3  +  6;r22  +  12a;22  +  833  10.  a;3-6a:2 2/  + 12^:2/2-8  2/3 

11.  a^a^  +  danx^y  +  dab^X7f  +  b^if 

12.  82:3  +  12a:3y4.6a;3y2  +  _j.3,^3  13.  a6_3a4a:2  +  3a2^_^.6 

14.  a^  +  da^xy+da^xUf  +  x^y^      15.  8a;9  +  12^«  +  6a;3  +  l 

16.  x^-6x^y^  +  12x^y^-8y^  17.  8 ^-6-362:41/2  + 54 a;2  2/4_ 272/6 

18.  a;«-9  a;5  2/ +  27x4  2/^-27  ir3  2/3      19.  x^y^-dx^y^  +  Sj^y^  +  x^y« 

20.  27a;«-108a:4  2/2  +  i44^2^4_642/6 

21.  x^  +  dx^y  +  z)  +  dx{y  +  zf  +  {y  +  zf 


Exercise  39. 

1.  0:2-2/2 

2.  a:6-2/« 

3.  a2^2_i 

4.  9m2-25w2 

5.  16a2a;2_9&2 

,  6.  4a:4^2_9^2,2 

7.  ar*  2/^-16  a:2/ 

8.  25  a;«  2/^-49^ 

9.  x^y^z^-i9 

10.  1442:8-252/4 

11.  (a  +  6)2-l 

12.   (x^  +  7/f-Z* 

13.  a^-y^ 

14.  16^:4-256 

15.  8l2:4_6252/4 

16.  a^a^-b^y^ 

17.  x^-Sx^  +  lQ 

18.  16a:4._8^2^2+2/4 

19.  a'^x*-2aH^x'^y^  +  b'^y^ 


ANSWERS.  325 

Exercise  40. 

1.  a:2  +  9a:+20  2.  a;*  +  7a:+10  3.  x^  +  Ux  +  SO 

4.  a;«  +  8a;  +  7  6.  x^  +  llx  +  24:  6.  a:2  +  lla;  +  18 

7.  a;«  +  14a-  +  48  8.  a;2  +  15a:  +  50  9.  4x^  +  Ux  +  12 

10.  9a;2  +  12j:  +  3  11.  25a:2  +  25j;  +  6  12.  a;«  +  5a:2/  +  6?/2 
13..a;«+a;-30                 14.  a;2  +  4a:-21  15.  x»+6a:-27 

16.  a:9-5a:-14  17.  a;2-a;-72  18.  a:2  +  6x-72 
19.  a:«-10a:  +  21             20.  a:*-14a;+48            21.  a;2-17^  +  70 
22.  4.T«-8a;-21        23.  9^2  +  6^y_8y2      24.  a2a;2-2a6a--3i2 
25.  a;«  +  (a  +  6)a:  +  a&                         26.  x^—(a—b)x—ab 

27.  a;'  +  (a— 6)a;— a6  28.  a:*— (a  +  &)a;+a6 

Exercise  41. 

1.  2^«  +  7.T  +  3  2.  2.£2  +  l4.^  +  20  3.  3a2  +  10a  +  8 

4.  8a2  +  14rt  +  6  5.  Ca2  +  22rt  +  20  6.  5a;2  +  38a;  +  21 

7.  6a;2  +  7a:y  +  22/'  8.  12a^  +  nab-\-2b* 

9.  2a:«-13a:  +  15  10.  8  ^2-30  a: +  7 

11.  Sx^-Uxy  +  lOy^  12.  6^:2-13 .xy  + 6 y* 
13.  2m«  +  mn-67i«                           14.  2a:4^4.^8_30 
15.  15x*-lla:2-12                          16.  2a^x^-ax-10 

17.  20a;«-71a;+63  18.  2'j*—4x^y^  +  2y^ 
19.  122r*-23a;«y  +  102/*                   20.  ISi^s  +  lGtc-lS 
21.  20x^y^  +  xy-30                        22.  15m4  +  4m2-35 

Exercise  42. 

1.  a:  +  y  2.  x'^+xy+rf  3.  .< ^ ^_ ^-s ^^  + .?: i/*  +  3/' 

4.  x*+x^y+x^y^  +  xy^  +  y*  5.  a;'  +  .r*2/'+a:3/^  +  y* 

6.  4a:«  +  6x  +  9  7.  42;2-2a:  +  l  8.  a^  +  3ab  +  9b* 

9.  l_a;+a;«-a:«+2;4  10.  27^-18x2  +  12i:-8 

11.  Not  divisible.    Why?  12.  x^+x^ y  +  x^y^+x^y^  +  xy*+y^ 

13.  x^—x*y+a^y^—x^y^  +  xy^—y^  14.  a:*+2:«2/'+3/* 

15.  a;*— a;'y«+2/*  16.  Not  divisible.     Whyf 

17.  a;*+2a:*y'  +  4y*  18.  Not  divisible.    Why? 

19.  16a;«»-8a;"  +  4a;'<>-2.7:5+l        20.  Not  divisible.    Why? 

21.  l-9a;«  +  81a:*  22.  23.  125a»-25rt»  +  5a-l 

24.  a»»-a»i«  +  a«6'«-a3  6i8  +  6«*  25.  a^^-a' b"^+b^^ 


ELEMENTARY  ALGEBRA. 

26.  «5  +  3fj4y  +  9«3y24.27«2  2/3  +  81a?/4+2432^6 

27.  a5_3a4y  +  9^3^2_27a-i2/3  4.8i^2/4_243^ 

29.  8a3-12a2J  +  18a^'2_27^;3        30.  a;4_4a;2y2  +  i6y4 
31.  4.i;4-6a;2  2/3  +  9/  32.  a;8-2 a:^ 2/*  + 4 a^ 2/^-8 a;^^/!^ +  102/^6 

33.  x^7f  +  x^y'^  +  .r'^y^  +  x^i/  34.  a^x^^—abx^f  +  b^i/"^ 

36.  «4_9  f^2  yi  +  81  y4  37.  64  ^2  ^,2_8  «  6  c  +  c^ 

38.  4:X^-G.i^7f  +  9y^ 

39.  ^8-2  a,-«  y^  +  4:  .i:^  ij^-8x^  y^^  + 16  2/" 

Exercise  43. 

1.  a{a  +  b)  2.  b{a—c)  3.  a:  (.?•  +  «?/) 

4.  a;(x2  +  3a;-2)  5.  3a(a-2  6  +  3^/2) 

6.  2a^x{l  +  2ax~Sa'^x^)  7.  6x.v(;r-^2/- 2:?^ 2/^^-3) 
8.  5a:2(2.r2  +  3x-4)  9.  7r2(l-2r+3?-2) 

10.  (a  +  &)(2a  +  3  6)  11.  {a-x){a—b) 

12.  (m  +  7i)(c  +  c?)  13.  12a3&2c(a6c-2^/2c3  +  3a) 
14.  5pq{2pq^  +  np'^q'^-4r)  15.  12 ^ / (2 2:^-3.^2/3  + 4 y") 

16.  {a'^-rb'^)i4c-od) 

Exercise  44. 

1.  (a  +  2  6)(a-2  6)  2.  (2a  +  5&)(2a-5&) 

3.  (32:+72/)(3x-72/)  4.  {ay  +  2){ay-2) 

5.  (4+2)  (4-2)  6.  (^+8)  (a:- 8) 

7.  (a;  2/ +  10)  (a;  2/- 10)  8.  (9 +  2)  (9-2) 
9-  (abc  +  Q){abc- 6)  10.  2/^ (.^-  +  z) (x—z) 

11.  (a2  +  22)(^^2)(a-2)  12.  (a^+b^){a^  +  b){a^-b) 

13.  (4a24.922)(2a  +  3  2)(2a-3  2) 

14.  (92/'''  +  1622)(32/  +  42)(3  2/-42) 

15.  x^y^{x  +  y)(x-y)  16.  (;?:3  +  2/2)(a:3_^2) 

17.  (25  +  22)  (5  +  2)  (5-2)  18.  (^4^.2^)(a;2  +  2/2)(a:  +  2/)(a;-2/) 

19.  (x^  +  2/2)  (a;3  +  y)  (x^-y) 

20.  (m4  +  M8)(^2^^4)(„j  +  ^i2)(,^_,^2) 

21.  {a  +  b  +  c){a  +  b—c)  22.  (a—x  +  y){a—x—y) 
23.  (w— ?i+l)(m— w— 1)  24.  (2  +  .'r  +  2/)(2— a;— 2/) 
25.  (c  +  a  +  &)(c— a— &)  26.  {c  +  a—b)(c—a  +  b) 
27.  (5a  +  a;— 2/)(5a— a:  +  2/)  28.  (4  +  2— a:)  (4— 2  + a;) 

29.  {l+x-y){l-x  +  y)  30.  (7  +  2a-  +  22/)(7-2a;-22/) 


ANSWJERS.  327 

Exercise  45. 

1.  {x-y)ix^+xy  +  i/)  2.  (x  +  y){x^-xy  +  9f) 

3.  (a-l)(a2  +  a+l)  4.  {a  +  l){a^-a+l) 

5.  (x-2)(^*  +  2j:  +  4)  6.  {x  +  2){x^—2x  +  4) 

7.  (2a  +  6)(4a2-2a6  +  J2)  8.  (2a-6)(4a2  +  2a6  +  ^;2) 

9.  (3a-2&)(9a2  +  6a6  +  4i2)         lo.  (8a  +  26)(9a2-Ga&  +  462) 
11.  (a«  +  l)(a*-a2  +  l)  12.  (a  +  l)(a-l)(a2_a+i)(rt2  +  a  +  l) 

13.  (a;«  +  2)(2:4-2a;2  +  4)  14.  {x^ -2)  (0^  +  2x^  +  4) 

15.  (a:+y)(a;*— a^'y  +  ^'y'— ^y^  +  y*) 

16.  {x-y){x*  +  x^y+x^if+xy^  +  y^) 

17.  (.r  +  y)  {x^—a^  y+x^  y^—x^  y^  +  x'^  y*—x  y^  +  y^) 

18.  (2  x-y^}  (16  2:^  +  8  x^  y^  +  4:  x^  i/  +  2xy^  +  y^) 

19.  20.  (2a:«-3y3)(4a.-4  +  6a;2//3  +  92/«) 

21.  22.  {x—y){x^  +  xy  +  ^f){x^  +  x^y^-\-y^) 

23.  x{x+\){x^-x-\-\)  24.  x^(x-2){x^-\-2x-\-4) 

25.  (a:y  +  2)(a;2  2^— a:i/£  +  22)  26.  

27.  (4  w*  +  5  n*)  (16  m4-20  m^  n^  +  25  n'*) 

28.  a:y(a:2/-l)(^2  3/2+a;2/  +  l) 

29.  x{x-[-y)  {x^-x y + y^)  {x^-x""  y^ + y'') 

30.  {x  +  y-z){{x-\-yf  +  {x-^y)z-\-z^ 

31.  (a;+2/  +  2){(a;+y)«-(x  +  2/)2  +  2*f 

32.  {x-y-z)\x^-k-x{y  +  z)  +  {:y  +  zf\ 

33.  (.r  +  7/  +  2)|a:2-a:(2/  +  £)  +  (2/  +  2)2f 

34.  (a  +  6_c_tZ)  \{a  +  bf-^{a  +  b){c-^d)-\-{c^-dy\ 

Exercise  46. 

\.{x-\-yf  2,  {x-zf  Z.  {x+\f  4.  (a--2)2 

5.  (.e  +  9)«  6.  (2a:-3)2  7.  (3a:  +  2y)2 

8.  (5x  +  l)2  9.  {x^-Qf  10.  (a;+2)2(x-2)« 
11.  {x  +  yf{x'*-xy  +  y'^f                   12.  {ah-cdf 

13.  (2a;+33/)«(2a:-32/)«  14.  

15.  (a;  +  l)«(j;*-a;3  +  a:'-a:+l)2  16.  x^y^(x*-\-2f 

17.  (a;«  +  y»)2(a:  +  2/)»(x-i/)«  18.  

19.  (2a;»  +  3)«  20.  <dx^(x^-2y^f 

21.  22.  y'^{x^  +  yzf 

23.  {x^-2yy  24.  16(a;«+2y«)«(a:«-22/«)« 


328  ELEMENTARY  ALGEBRA.      . 

Exercise  47. 

1.  (a:  +  3)(a:  +  5)  2.  {x  +  4){x+l)  3.  (a:  +  4)(a;  +  2) 

4.  (a-4)(a-3)  5.  (a-7)(a-2)  6.  (a-8)(a-5) 

7.  (a:  +  5)(^-3)  8.  (:c+7)(a;-4)  9.  (a;  +  8)(a:-2) 

10.  {:x-b){x+\)  11.  (:i;_7)(a;  +  3)  12.  (a:-10)(a:  +  8) 

13.  (2a;  +  3)(2.2;+l)        14.  (22;  +  4)  (2a;  +  3)        15.  (3a;+2)  (3a;  +  l) 

16.  (a;  +  3a)(x  +  a)  17.  (a;-5a)(^+3a)       18.  

19.  {^y  +  2z){^y-z)                         20.  {Qx  +  bh){Qx-h) 
21.  (2aa;— 5)(2aa;  +  3)  22.  

23.  {:x'^-4:a){x^-Za)  24.  

Exercise  48. 

1.  (2x  +  3)(:r+l)  2.  (2 a;  +  3) (a:  +  4)  3.  (3a:+2)(22:+l) 

4.  (32;  +  l)(2a;  +  3)  5.  (2'rc-3)(a;-2)  6.  (2a;-3)(x+2) 

7.  (2a;+5)(a:-3)  8.  (4a:-3)(3^+5)  9.  (3aH-5)(2a-3) 

10.  5(3a  +  7)(a-l)  11 12.  (2a-Z<)(a  +  6) 

13.  {2a^-b){a  +  2h)  14.  (3.2;-?/) (2 a:- 3 2/) 

15.  (3w-.2i')(2?/  +  3v)  16.  

17.  (3a;-^  +  5)(2:^;2_7)  18.  (2  a  6-3)  (a  &  + 2) 

Exercise  49. 

1.  {:c^^x  +  l){x^-x  +  r)  2.  (x2  +  2a:  +  4)(a;2-2.r  +  4) 

3.  (a2  +  a&  +  62)(a2_a&  +  52) 

4.  (a2  +  a  c  +  c2) (a2-a  c  +  c^)  {a'^-a^  c^  +  c*) 

5.  (4a2  +  2a  +  l)(4a2-2a  +  l) 

6.  (a4+2a2j  +  462)(a4_2a2^,^.4j2) 

7.  (a:4  +  a;2  2^  +  y2)(a;4_^2  2^  +  y2)  s.  (a:4  +  ^2y3  +  ,^6)^^_a;2^3  +  2/6) 
9.  10.  (9x^  +  Qxy  +  4:y^){9x^—Qxy  +  4:y^) 

11.  a-2  +  a;  2/2  +  2/4)  (a.2_^  y2  ^  2/4)  (pA_^i  y4  ^  y8^ 

12.  {a^x^  +  axy+y^){a^x^  +  axy  +  y^) 

13.  a'^y'^iy^  +  ay  +  a^){y^—ay  +  a^) 

14.  (a4  6*  +  2  a2  ^,2  +  4)  (^4  ^,4_ 2  ^2  62  ^  4) 

15.  {9  +  Sb  +  b^){d-?jb  +  b'') 

16.  (25  +  5a;2/+a:2  2/2)(25-5;cy  +  a;2  2/2) 

17.  {4+2  z  +  z^)i4:-2z  +  z^)  (16-4:2^+2^) 

18.  y*(a;2+a;2+22)(a;2_a;2+^'^) 


ANSWFRS.  329 

Exercise  50. 

1.  {a  +  b){x  +  ij)  2.  {b  +  c){x-y) 

3.  {a-b)ix-z)  4.  (&  +  3)(a  +  2) 

5.  (3-2/)(3  +  .r)  6.  {a  +  2b){2x  +  dy) 
7.  (2a-db){dx  +  2y)  8.  (aa;  +  3)(6a:+2) 

9.  (a-6)(a:y+6)  10.  {a  +  b){a-b){x^  +  if) 

11.  (a  +  6  +  c)(a  +  &-c)  12.  {x  +  y-l)(x-y-{-l) 

13.  (a;+2/+l)(^-2/-l)  14.  (2a:+2/  +  ^)(2a-  +  .y-2) 

15.  2z+2x+l){2z-2x-l)  16.  (a  +  6  +  4)(a  +  &-4) 

17.  {5+x  +  a)i5-x-a)  18.  (^2+a:-l)(a:2-a:  +  l) 
19.  (x-2  +  7/2  +  2/)  (.x2  +  ?/2-y)  20.  (m2  +_p2  +  ^)  (m2-^2-5) 

Exercise  51. 

1.  ab{a  +  b)(a-b)  2.  3a(«  +  2)(a-2) 

3.  2a(a-&)(a«  +  a6  +  &«)  4.  3&(a  +  6)(a2-a6  +  &2) 

6.  xy(x^  +  y^){x  +  y){x-y)  6.  «a:2(a;+l)(a;4-a;«  +  a;«-rr+l) 

7.  a  6 (a-b)  {a  +  6)  (a^-a  b  +  i'^)  (a^  +  a  6  +  &«) 

8.  2a«6«(a«  +  6*)(a4-a»62  +  64)  g.  5(a  +  2/)2 
10.  2ac{a+Sy                                 11.  a;2(a:i/-l)2 

12.  3aa;(x  +  3/)«(a:-2/)«  13.  4y(y  +  lf(y^-y  +  l)^ 

14.  2  a;  (a; +2)  (a; +3)  15.  xy{x-4){x-o) 

16.  4a&(a+7)(a— 6)  17.  a(a  +  6  +  c)(a— 6— c) 

18.  a^a  +  b  +  c){a^-ab-ac  +  b^  +  2bc  +  c^) 

19.  ac(a-2c)(a  +  c)  20.  4(a  +  3)(&  +  2) 
21.  y{a-b)(x-y)                             22.  (a;-2/)(a:  +  y-l) 

23.  {x  +  y){x  +  y—l)  24.  a;«(a  +  &  +  c)(a  +  6-c) 

25.  x{x—yA-z){x—y—z)  26.  {x—yf 

27.  a;(a:+2)»  28.  (a^  j/«+l)(a3/  +  l)(ay-l) 

29.  (a;2/-2)(a:2  2/«  +  rr 2/2  +  22)  30.  {x^  +  y^){a^—x^y^-\-y^) 

31.  (i;  + 1/)  {x-y)  (a;«  +  3/«)  (a;«-a:  y  +  y*)  (a;« + a:  y + y«)  (a:*-a;«  y« + y*) 

32.  ?n(mn2  +  l)(mw«-l)(m*n4+l)  33.  (lla«  +  126V 
34.  (4a  +  3&)(4a-6)                         35.  (mn-2)(3-a) 

36.  (a  +  &  +  l)(a-6  +  l)  37.  {ab-\-a  +  b){ab—a-b) 

38.  (l+x+y)(l— a;— 2/)  39.  (a  +  w— n)(a— m  +  w) 

40.  (l_a-6)(l+a  +  ft  +  a«  +  2a&  +  6«) 

41.  (a4-a«6«  +  &4)(a2  +  a6  +  i2j(a2-a6  +  62)  42.  2  a;  x  2  3/ 


330  ELEMENTARY  ALGEBRA. 

43.  a^  (a  +  y)  (a^—a^  y  +  a'^  y^—a^  y^  +  a^  y'^—a  1/  + 1/) 

44.  (a  +  j_l)2  45,  —t(2x  +  y  +  z)(y  +  i^ 
46.  {x+y  +  2)  (x—y—z)  {x^  +  y^  +  2  y  z+ z^) 

41.  {x+yf  4Q,  (\  +  a-b)(l-a  +  b) 


Exercise  52. 

1.  4:xy 

2.  5a2 

3.  10a:2/2 

4.  5a2&2 

5.  6m2w2 

6.  {a  +  hf 

7.  3(a;  +  2/)« 

8.  m{m  +  n) 

9.  2aa;(a;2  +  2/2) 

10.  2(m-?0^ 

11.  y{a-hf 

12.  (a  +  5)(a_J) 

13.  2{x-7jf 

14.  3  a  (m-7i)2 

15.  {m  +  n){m—n) 

16.  (p-q){a-b) 

17.  (a-&)2,  or  Q)-af 
Exercise  53. 

18.  ±(a-6) 

1.  a  +  6 

2.  a-{-h 

3.  a-b 

4.  a  +  & 

5.  a:;— 2/ 

6.  x  +  y 

7.  a:—?/ 

8.  a;  +  y 

9.  a;-4 

10.  a^  +  2a 

11.  a:2_8a;+i5 

12.  x^-y^ 

13.  a:  +  i/+s 

14.  a:+3 

15.  a;+3 

16.  a:  +  a 

17.  m  +  n 

18.   X^  +  7f 

19.  x^  +  2y 

20.  a:  +  2/  +  2 

21.  a:  +  y+2 

22.  a;+2/ 

23.  a;(a;2  4-7) 

24.  2 2;+ 6 

25.  x^+xy  +  y^ 

26.  x^+Zx 

27.  x^  +  4: 

28.  .-r^-a^ 

29.  a;2  +  :ry  +  2/' 
Exercise  54. 

1.  60a;2/ 

2.  72  a2  53^3 

3.  96a&ca;«?/2 

4.  132^:2  2/3  22 

5.    144a2J3a;3  23 

6.  1008  m^n'^x^ 

7.  (a +  6)4 

8.  100(rc+2/)5 

9.  a^{x-yf 

10.    24a2&2(^  +  ^)4  11,    36:^3  23(3^2 +  2/2)5 

12.  {a  +  b){a—b){a^-^¥) 

Exercise  55. 

1.  {a-\-bf{a-b)  2.  {a-bf{a^  +  ¥){a  +  b)  3.  x^-y* 

4.  (iK-2/)(a;+y)(a;2^.a:^  +  2/^)  5.  (x—y){x-\-yf 

6.  (a; + 2/)  (a:-2/)2  7.  {x + y)  (a;-2/)2  (^2  +  a:  ^ + y'i) 

8.  (a: 4- a) (a; +  6) (a; +  c)  9.  abc(a—b) 


ANSWERS.  331 

10.  {a-b){b-c)  11.  {x+yf(x-y)(x^-xy  +  y^) 

12.  {x  +  3)(x-i)(x+2)  13.  (x-l){x+4){x-5) 

14.  {a  +  b)im-\-7i)(p  +  q)  15.  (a— &)(a  +  6)(j:-^) 

16.  -{a  +  b  +  c)  {a—b—c)  (b—a—c)  (c—a—b) 

17.  (a  +  &)3(rt-6)  18.  (.i+2/)(a:-3/)(a:*+a;»2/'  +  2/^) 
19.  {2x+5)ix  +  'd){x-2)  20.  (3x  +  2)(22:  +  3)(2a:-3) 
21.  axyix^-y^){x*+x^y^  +  y^)  22.  ;cy (.f  +  2)(a:  +  3)(a;-3) 
-23.  a^  +  x2?/2  +  ?/4  24.  .t6  +  2/* 


Exercise  56. 

1.  1100 

2. 

540                   3.  900 

4.  700 

5.  700 

6. 

1728                 7.  4 

8.  5 

9.  4 

10. 

27                   11.  20 

12.  7 

13.  (a;+y)» 

14.  x^-y^ 

15. 

x'+x'y'+y^ 

16.  x^+4x 

+  4 

17.  rr  +  4 

18. 

1 

19.  (a-^»)2- 

-c« 

20.  36 

21. 

-2 

o 
22.  -3 

23.  280 

24. 

0 

Exercise  57. 

1.  x=4,  y=r),  z=  6  2.  a:=2,  2/=3,  z=l 

3.  a;=3,  y=5,  2=  4  4.  ic=5,  2^=4,  2=3 

5.  x=S,  y=A,  2=  5  6.  x—\,  y=—\,  z=0 

7.  x=2,  y=S,  2=  5  8.  x—o,  y=4,  z=S 
9.  x=7,  y=5,  £=  1  10.  a;=4,  2/=5,  2=6 

11.  a;=6,  y=7,  2=10  12.  x=l,  y=2,  2=3,  m=4 

Exercise  58. 

1.  10,  20,  GO  2.  20,  30,  40  3.  $1900,   $300,   $1300 

4.  125  A.,   150  A.,  225  A.  5.  GO  ct.,  40  ct,  80  ct. 

6.  $100,   $G0,  $8  7.  15,  6,  9 

8.  10  yr.,  20  yr.,  30  yr.  9.  $100,   $G0,  $20 

Exercise  59. 

1  ^  2  IL         <?  -^        A  iy^* 

3a  4ac«  3^«2«  5a;» 

^         3  g^  b{j^-y^)  ^    _1_  ^    a  +  x 


4{x-\-y)  a  a—b  a—x 


332  ELEMENTARY  ALGEBRA. 

a;  +  2  ■  x  +  '^y  '  2x-'6y  x^+y"^ 

^^   x^-xy  +  y^  ^^   2x-5y  ^^   x-y-z 

x+y  '  2x+oy  '        x 

16.  thi±£  1,.  5±|  18.  ^1=5 

IQ    ^:i5  on    4a2  +  6a&  +  95g  o^^  +  a^rg^  +  a^ 

a;2-2/2  "*•  x^+xf^y^+x^y^+y^  a-b 

Exercise  60. 


1. 

X 

2.  '2/-^ 

3. 

ax+x+a 

X 

4. 

n 

X 

5.  — 

a  +  a; 

6. 

2x 
x—a 

7. 

a  +  ;r 

a:  +  a 

9. 

2y^ 
x^+y^ 

LO. 

4a  +  3 

26 

Exercise  61. 

12. 

a;2-33 
x—4: 

1. 

2. 

a 3.  a  +  x  + 

x 

1 

X 

4.  x  +  l-\ 

x-1 

2  ifi  2  v^  v^ 

b.x—y-\-^^—  6.  x^—xy  +  y^ ^—  t,  x+ ~ — 

^      x+y  ^     ^      x+y  x+y 

8.  3a;  +  5 -r  S.x^-xy  +  y"^  lo.  x^  +  xy+y^+ -^ 

11.2x4-1-77-^  12.  5a:  +  6+     ^^ 


2^-1  *'"  "-^^""^Src-e 

13.  2a;2  +  3a:-l+  ^-^^  14.  ^x'-4.x  +  n+  ^=i? 

2a;  ^        2a;+l 


Exercise  62. 

1     ^^       &y       CTg  a2:  +  &2r     ay—hy      bx 

abc^  abc''   abc  '     xyz    '      xyz    '  0:3/2; 

_     a^x      ¥x      cxy  ^    d^—ab     ab  +  ¥         c 

O'    — 1— »     — T-,     —r^  4. 


a62/'   a&y'  a&y  a2_j2  '    (j2_^,2  ^  a2_^,2 

(a+a:)2     (ct— .-r)^     a^+x^  a^—a^x  +  ax^         b         ac  +  cx 

'     a2_^2'      ft2_^2'     a2_a;2  O-  ^Sljr^i  '     ^3^^'     "^sTf:^ 


7. 

8. 

9. 

10. 


ANSWBJiS.  333 

;a:— 15  5a;+15 


{X  +  2)  (2; + 3)  (x-5) '  (X  +  2)  (« + 3)  {x-5) 

x—a  +  b  x  +  a—b 


{x+a+b)(x+a—b){x—a  +  by  (x  +  a+b)  (x+a—b)  {x—a+b) 

2x-6  Sx—S  4a;-8 

{x-l){x-2){x-'S)'  ix-l)(x-2)(x-'6)'  {x-l){x-2){x-d) 

X         4a:*— 4:r+l     4a;'^  +  4a-+l 
4  a:*-!'        4j;2-1      '       4a:2-l 


o+aa;         h  +  bx        c  3.r— 6     4a:— 8 

12.    — 5-, :; -T  ,     z        Ti  lo 


14. 


l_a;2  '         l-a;2'   l-a;'  (a:-2)2'   (a:-2)«'   {x-^2y^ 

S—x  2-2x  6— 3a; 


15.  - 


{x-\) (x-2)  {x-d) '  {X- 1) (x-2) (a:-3) '  {x- 1) (a;-2) (a;-3) 
ab—ax  bc—bx  ac—cx 


{a—x){b—x){c—xy       {a—x){b—x){c—x)        (a—x){b—x){c—x) 


Exercise  63. 

2                                2  a  a;  a^b^  +  a^c^  +  b^c^ 

1.  ^ 5  2. 5 5  o.  J 

1— a*                             a^—x^  abc 

b  +  c—a                   ay— ax  2ax^  m  +  n 

4.  7 0.  O.  —7 i  7.  

a6c                          a:y  x^—yr  m—n 


z.4^.       9.4^        10.^  x,.,,^^-i=^ 

12.  j5«^  +  2fty  13. -^—^-^^ 

14.^-1"  +  ^  15.2=^  16.46-a-2l+*! 

x       y       z                              X  abx 

*  6  a:      12  y      4  s                            '15  a;  40  y      12  2 

,^   2a»  +  2aa;«  ^^      2a:«  ^,      a;«y 

19.  -7-^ ^z^  20.  ^ 5  21.         ^ 


(a2-a;«)«  a:8  +  y3  {x+yf 

22.  , T^^ TT,  23.  0  24.  , ,,     ^^     ,, ^ 

{x-\){:x-'6)  (x^y  +  z){x-y-z)(x+y-z) 

26.     ,,  ^,    ,,  26.  -is  27.  0 

a;(l— 4a;*)  a;+2 

Exercise  64. 

a*bx  n   ^^^  ^   ^^^  A    x^  +  .V' 

1.    2.    J —  O.    — Ya'  4.    — — ^~ 

c  (i  d^  xy 

a;«+/  °-  a»-a6  +  6*  a-6  a;+2 


334-  ELE3IENTARY  ALGEBRA. 

9.  —  10.  -—5  11.  ;^  -^  12.  ^ 

a;  c'a  dm  71  xy 

lO.    14.     rrr 10.    ;v 

X — y  Vlmn  ic+o 

16.  , V^  17.  (^^=25)^)  18.  ^'-<°  +  f 

19  ^y^y^^)  20  ^~^~^  21  ^'^+-^y+y' 

■  2(a;2— ^2)  '     abc  '        x+y 


1. 


bdy' 
4.  ax{a—x) 


20. 

abc 

22. 

x^—xy  +  y^ 

x-y 

Exercise  65. 

2. 

xy 
bc^z^ 

5. 

cxy 
bd^ 

acx^  ^      xy  '       ahx^ 


{c  +  df 


cd 


adx  e  ^         x{a—x) 

8. 


'  bc^y  '  c{a~x)  '  a{a^—ax+x^ 

x^y^  +  y^  x^-1  {a  +  b){x+y) 

16-  'V'!"^f  17.^     .  18.1 


Exercise  66. 

ay                       a^  {x  +  lf  ac  +  b 

bx                       y^  (x-l)^  '  ac-b 

a:                        x^  +  ^x+Q  a  g    _«_ 

■  x—a                  '  x^—bx+Q  '  x{a—x)  '  x+1 

a;2  +  a                    a;-6  x^  +  2x  a-5 


Exercise  67* 

,    JL  n    ^  o    16;^^V  J1024 

^"  216  <^4  '='■     81  ^8  ^-  59049 

{x  +  yf  q'{q-lf  '{■»^  +  yf  '  {m  +  nf 


ANSWERS. 

.335 

Exercise 

6S 

1. 

^'  c  +  \^d 

A         2.rV 

5.     -                  6.  \—x—- 

y                                 1+^ 

7.  0 

8.  1 

^^'  a«  +  (5»2 

11.  t^" 

2a  6 

12.  V                  14.  -T — , 

15.^;+^+ 

y*    y 

1 

X 

^y' 

rr»      2rr?/      v«     a^     ^     ?/» 
^^'a^^  a6  ^62'  y^     "+ a« 

,„    ,      3a      3a«      a^    a^      36*-« 

18    ^*'      ''^ 

19. 

--h 

20.  a;«  +  2j;  +-  +  -+  3;  x^-2x-\+  -  +  -« 

X         X^  XX* 

«,    /a+rr\2     /a— a;\*        8a3a:  +  8aa:» 

^^   a^     ay       -  ^«   a*     .,      J' 

a3      o^^      0^2      ^•.   «'_«^      ^_^ 
'  ic^      a;*//      aw/*      2/*'  X*      a;*y      xy*      y* 

-(-3(-f)(>-f)^(-4:)(«4)K)= 

/x     ^\(^     "\ 

(«-f)(-f)(«'-y-3(«'-if%") 

Vy     xJ\y^  a;2/'  Vwi*    7i^/\m*    m^n^    7i*J 

29.  a:4-a;'  +  l-  ^,  +  ^  30.  | 


336  ELEMENTARY  ALGEBRA, 

Exercise  69. 

1.  a;=3  2.  x=4:  3.  x=\  4.  x^l'Z 

5.  2^=24  6.  x=U  7.  x=^%  8.  x=.'^X 

o  2 

9.  x=0  10.  x=7  11.  x=3  12.  x=7 

14  5  4 

13.  ic=ljx  14.  a;=-y  15.  a;=8  16.  a;=-y 

22.  x=-3  —  ^_     ..3,^  -^.  ^_„  ^ 

4  82  5 


5  5 

17.  x=l-i^  18.  a;=^  19.  x=8  20.  a;=3 

21.  a;=-r  22.  a:=-3  23.  a;=-l;^  24.  a;=5^ 


Exercise  70. 

1.  x= 1  2.  x=l  3.  x=c—d 

a  +  b 

,       „  _            «c  ^          c+bm—an 

4.  a;=:m2  +  ?i2  5.  a;=-; 6.  a:= — — 

b—c  m—n 

_       12  o      _     2ffl  +  &  _m{m  +  n) 

18— 17  a  9  {m—nf 

&2_a2  aH-cd^  ,^            ar^-c2 

10.  a:=:    ^  ■  11.  rr=— -T- 12.  a:=-— , 

2b  ad+ac  md—nc 

_-                 a&c  .^  bcd^ 

13.  a;=— 5 5—  14.  a:= 


ab  +  ac  +  bc  acd  +  b^d—bc^ 

,_             1                       ,_          {a—b)c  __  cd—ab 

15.  a:= r  16.  2;=^^ r^^  17.  x=- 


a  +  b  ab  a  +  b—c—d 


Exercise  71. 

1.  a:=ll  2.  a;=f~  3.  a;=25a  +  24J 

06 

2  _  ,  «  457  „  a  6-1 

4.  a:=— —  5.  a;=4  6.  ^=-7^^  7.  x= 


9  102  -bm  +  ii 

8.  a;=2  9.  x=3  10.  a^^:^  11.  x=4.-r 

19  4 

11  17 

12.  x=l-^  13.  a;=-=-a  14.  x=:^ — -  15.  x=zr7i 

2  7  2— a  10 

16.  a;=-4  17.  a;=0  18.  a:=~  19.  a;=4 

16  2  c 

20.  x=——-        21.  0;=-'^ 7—  22.  a:=.8 

a  a  +  o 

4a62_l0a  «^  «  «^  o  1 

23.  a;=  — . ^^—  24.  a;=3  25.  x=B  ^ 

4a— 3 0  2 


ANSWERS.  337 

Exercise  72. 


1. 

10 

2. 

$5,  $50, 

$200 

3. 

.  18 

4.  80  yr. 

5. 

300  bu., 

200  bu. 

,  IGO  bu. 

6. 

< 

7.  24,  60 

8. 

$80, 

$r 

rs   9. 

28 
87 

10. 

$2000, 

$9000, 

$5000 

11.  72,  91     12.  90  yd.,  75  yd.,  35  yd.      13.  400 

14-  $00       15.  $20,000     16.  196  lb.     17.  84 

18.  $162,  $118,  $104         19.  20  yr.,  16  yr. 

20.  $3000,  $2500     21.  300  bu.,  200  bu.     22.  84,  96 

23.  GO  ct.      24.  66      25.  $75     26.  $12,500,  $10,000 

27.  $133  4     28.  $1000.  $700    29.  $2.91 1   30.  30^ 

31.  14^^     32.  7^^      33.  87^^       34.20^ 

35.  $5000  36.  $800  37.  $500,  $300         38.  $720 

2  200 

39.  5^  yr.  40.  15  yr.  41.  —  yr.  42.  G% 

100  2 

43.  ^^%  44.  677^  45.  $75  46.  Q% 

n  0 

4n.  $100  48.  75^  49.  $4800  50.  22  ^  mi. 

61.  24  mi.  52.  2  mi.  53.  I5?  da.  54.  13  ^   hr. 

37  o 

56.  17^  hr.         56.  2  da.  57.  $300  58.  $800,   $500 

59.  $200,  $280  60.  1740  61.  15  yr.  62.  $900 

63.  15  lb.  64.  $60  65.  6^  hr.  P.  m.  66.  2  p.  m. 

67.  29—  min.  past  4;  5:J^  min.,  or  38 tt-  min.  past  4;  54-j-  min. 
past  4.  gg^  ^g  ^^  ^,j.^  gg_  2Q  ^^^  rjQ^  ^c)  ct. 


Exercise  73. 

1.  a:=3,  y=l  2.  x=4,  y=\  3.  x=o,  y=5 

4.  a;=-,  y=-  5.  x=6,  y=5  6.  x=5,  y=A 

7.  x=i,  y=S  8.  x=-2,  y=-S  9.  x=-l,  y=0 

10.  x=5,  y=4  11.  a;=4,  3/= -4  12.  x=2^,  y=^'^ 


338  ELEMENTARY  ALGEBRA. 

Zxercise  74. 

1.  ir=4,  2/=3  2.  3—2,  y=-2  3.  a:=4,  2/=4 

4.  a;=-5,   2/-2  5.  r^=7,   2/=2  6.  a;=:3,   2/=10 

12  11 

7.  a;=l^,   2/=i7  8.  a;=^,   y=  ^  9.  rr=3,  3/==5 

1  1 

10.  x=^,  y=z~ 

Exercise    75. 

1.  a;rrl4,   y—U  2.  x=10,  y=12  3.  a:=10,   2/=3 

4.  a;=3G,   .^=90  5.  a;=13,  y=17  6.  a;=5,^,  y=4~ 

lo  13 

7.  a;=9,  y=lo  8.  a:=-60,  2/=5  9.  a:=10,  y=2^ 

o  j^ 

10.  x=TA-,  y=m^  11.  a^=G,   7/=-9 

100  40  o         .-  ^ 

12.  ^=-j^,   y=;32i  13.  a:=la,   2/=8 

14-  a:=:-27,   ?/=13  15.  a:=3,   2/=2  16.  x=^,  y=-5 

Exercise  76. 

2.  x=2,  y=3  3.  a;=4,  2/=6 


1       _1. 

2'  ^~3 


11  oil 


Exercise  77. 

c+d  c—d  ^  n—hm  am—n 


2-  ^=-77— IT'   2/= 


2a  '   -^       26  a-6   '   -^        a-& 

2^        ^'   "^     2  ^        ^  ms—nr'  ^     nr—ms 

an—hm  bm—nn         ^  c  +  n  bn—ac 


cn—bd''   "^      mc  —  ad  a  +  b^   ^      m{a  +  b) 

an+b  b—am 

7.  a;= ,   y= 8.  x=l,  y=0 

m+n  m+n 

^  b  71— bd  bm—bc 


nc—md'  ^     nc—d 


m 


10.  a;= 7 /     ,   y=        ^ 


mn{c7i,—dm)'   ^      7n7i{cm—d7i) 


ANSWERS.  339 

Exercise  78. 


1.  $168, 

$175 

2.  $()00,  $400       3.  4,  3 

4  ^  ^ 
*•  IG'  8 

5.  CO  bu.,  40  bu.    6.  $100,  $20 

"■l 

B-l 

9.  8,  3      10.  34      11.  39 

12.  12 

13.  24 

14.  12  4  da.,  21  i  da. 

15.  8  da. 

16. 

15-^  da.,  17  da.      17.  42  da.,  C3  da. 

18.  10,   18  19.  16  ft,   10  ft.  20.  15  rd.,   9  rd. 

21.  15  yd.,   10  yd.        22.  16  rd.,   10  rd.         23.  45  in.,"  03  in. 
24.  150  yd.,  30  yd.  per  min.,  20  yd  per  min.  25.  $3 

26.  8  ft.,  6  ft.  27.  3500  28.  8  persons,  50  s. 

29.  32  ft.,   21 1^  ft.,   691  i  sq.  ft.  30.  $600,   6% 

31.  $800,  5  yr.  32.  25  mi.,   15  mi. 

33.  15  mi.,  5  mi.  34.  $500,   $400 

Exercise  79. 

1.  x=2,  2/=60,  2=26  2.  x=S,  ?/=5,  2=4 

3.  x=2,  y=3,  z=\  '      4.  .t=12,  y=U,  2=36 


1       1       1 

6.  x=^,   y=3,  z=-^ 

,111 
6.  x=^.  y=^,   2=^ 

m  +  n—r          m—7i  +  r 

n  +  r—m 
'=       2c 

8.  a;=4,  y=5,  2=6 

9.  ;r=10,  2/=8,  2=6 

10.  a:=8,  y=16,  2=24 

11.  x=l,  y=-l,   2=0 

12.  x=ry,  y=4,  2=3,  u=2  13.  x=-^,  y=  ^,  z=  ^ 

14.  j:=12,  y=24,  2=36  15.  x=a,  y=b,  z=c 


Exercise  80. 

1.  00  ct,  35  ct.,  65  ct.  2.  100  A.,  90  A.,   120  A. 

3.  $600,   $400,   $200  4.  10  da.,   20  da.,   30  da. 

13  1 

5    20  yr.,   35  yr.,   42  yr.  ^'  '^ij  ^*''   ^13  ^^''   ^'^7  ^^ 

7.  20  hr.,  30  hr.,  40  hr.       8.  $252  ^  $200,  $47  ^ 
9.  346  10.  $600,  $800,  $1000 


340  ELEMENTARY  ALGEBRA. 


11.  $450,   $225,   $237|,    $87|-      12.  2  ct,  3  ct,  5  ct. 
13.  $40,   $60,  $80  14.  5  gal.,  3  gal.,  2  gal. 


Exercise  81. 

1.  14,  6  2.  36  yr.,   12  yr.  3.  $90,  $80 

4.  3^  da.  5.  $700,   $300,   $800  6.  40 

7.  5  8.  120  A.,  160  A.  9.  $4,   $1 

10.  i(c  +  ^),  \{c-d),  22  yr.,   14  yr.  12.  5,  6,  7 

,,    100— flfc?     ac— 100     _    „  ,_    d—am     d—an     ,,     ^^ 

11.  -p-,   -j—;  6,   8  13.  -,   ;   14,   10 

c—d  c—d  n—m       n—rn  ' 

,^    bn—dm     cm— an     ^^^    ^„^       ,_     ad  ad         ^.     ^^ 

14.  -T :j,  -t tt;   $25,  $35       15.  -^ — ,  -^ — ;   50,   75 

be— ad      be— ad  d—c     a—d  +  c 

,^    bd—ac—ab{b—a)     be—ad—ab{b—a)      ..     .. 

1^- p=^r- — ' pz^2 ;  14,  10 

,_    lla+&     _-,                        __    ce  +  bd     be— ad     ._     .. 
17.  — ^ — ,   85  18.  r^,  j^;   20,   10 

2      '  ae  +  b^ '    ae  +  b^  '       ' 

19.  ^(a  +  b-c),  ^(a  +  c-b),  ^{b  +  c-a);  40,  54,  36 

63(aQ?-5c)     6S(ad-bc)^ 
^°-    63^-17  6  '     17a-6ac  '■    ^'  ^^ 


Exercise  82. 

1.  c'^  +  4:C^d  +  Gc^d^  +  4:ed^  +  d^ 

2.  a'^-4:a^d  +  6a^d^-4ad'^  +  d* 

3.  a;5  +  5a?*2/  +  10a;3  2/2+10a;2  2/^  +  5;r2/^+?/5 

4.  x^-5x^z +  10 x^  2^-10  x^z^  +  nxz'^-z^ 

6.  7n^  +  6m5n  +  15m*n2  +  20m3  7i3  +  15m2n*  +  6mw*  +  w^ 

6.  m®— 6  TO^  n  + 15  m*  /i^— 20  m^  w^  + 15  m^  w^— 6  m  n^  +  n^ 

7.  c'-7c«a;  +  21c5x2-35c4.'r3  +  35c3a;4-21c2a:6  +  7ca;8-a;' 

8.  x^  +  ^x'^z  +  %%x^z'^  +  mx^z^->t'70x^z^  +  bQx^z^-\-2Sx^z^-¥%xz'^+z^ 

9.  a;8-8a;'^  +  28a;V-56xV  +  Wa:*2/^-56a;V  +  28a;V-8a:2/'  +  2/« 

10.  c9  + 9 c8 2  +  36 c' 2^  +  84 66^3  + 126 c5^4  + 126 c4  25  +  84c3 ^6  + 

36c2  0'  +  9c28  +  29 

11.  yo- 10 a: 2/»  +  45 0:2 2/8_  120 ^:3y7  + 210 a.-4y6-252a;5y5 +  210.^6^4 

- 1 20  a;' /  +  45  0:8  2/2  _  10  a:9  y  +  a;io 

12.  2"  + 11 2^0  y  +  55  ^9  i/2  + 165  28  ?/3  +  330  ^t  _,y4  +  4(32  ^6  ^5  +  4^2  z^  y^ 

+  330  £4  2/7^.165  23^8  +  5522^  +  112  2/10  +  ^1 


ANSWERS.  341 

17.  x*  +  ij:^  +  Qx^  +  4x  +  l  18.  a^-4.x^+Gx^-4x+l 

19.  a:*  +  5u;4  +  10a;»+10a;«  +  5a:+l 

20.  a;'-5a;*  +  10a:«-10a:«  +  5a;— 1 

21.  l  +  6z  +  loz^  +  20z^  +  15z*  +  Qz^  +  z* 

22.  1-62  +  1522-2023+1524-0^5  +  26 

Exercise  83. 

1.  a*  +  8an  +  2Aan^  +  'S2ab^  +  16b* 

2.  81a4-21G«3^  +  216a2  62-96a63  +  i664 

3.  32a:5  +  80^^  +  80a;«  +  40a:2  +  10a:  +  l 

,      ,      Sa^x      10a3a:2      lOa^a;'      Saa;*      rc» 

4.  o^  —  —  + s s —  +  — 7 J 

a*      5a<c      10«»c«      10a''c»      Sac^      c^ 
°"  ft*^"^  ^M  "^    63(i*    "•■    6*d»    "*■   bd*  "'"^6 

_5      10_10       5^_^ 
a:*  "^  a;'"      a;''  "*"  a;««     a;« 

7.  l-10a:«  +  40a;*-80a^+80a:8-32a;'o 

8.  a:'8  +  6a:"2/3  +  i5a.i2y6  +  20a;»y»  +  15a:«2/^2  +  6a-3y8  +  yi8 

9.  a'8-6  a»5  ^/4 ^.  15  «i2j8_20a9&>2  + 15  a«  616-6  a3  ^'^0^.^24 

lO..--12.-  +  60^-1604-^-^  +  ^ 

81 6^       Oi*""''         a*  "^  16  a* 

^_200   5     500         625         3125   ^     3125    ,^, 

243      243  ^  "^  243  ^     243  ^  "^  1944  ^  ~  7776  ^ 

13.  64  ai«  +  576  a'o  a;3 +  21 60  a8a;«  + 4320  a«a:»  + 4860  a*  a:'2  + 

2916  a«  .«»«•  + 729  a;'8 

14.  a:i»-6a:'<'2/*+15j*y»-20a:6^«+15a:4  2/^-6a;2yo  +  yi2 

15.  -32 a:«-240x*y-720a:«2/«-1080 a:* 2/3_8lOa: 3/4-243 3/5 

16.  ^^-^^y  +  6a:^y'-9a:3/3+^y4 

17.  flJo ^io_5a6a4+ 10  a«a:«- 4^  +  -.-.-      ^ 


a«a:«      a'®a;'« 

810  a8      243 


18.  32x'0-240a«a;'  +  720a4a.4-1080a«a;  + 

X 


Exercise  84. 

1.  x^-\-y^  +  z^  +  2xy  +  2xz-i'2yz 

2.  a:»  +  y«  +  2«— 2a-?/— 2a:2  +  2y2 


16 


342  ELEMENTARY  ALGEBRA. 

3.  x^  +  y^  +1  +  2 X 7j  +  2 x  +  2 y  4.  a'^  +  ¥  +  i-2ab  +  4:a~4b 

5.  a''  +  2aH  +  dan^  +  2ah^  +  b^ 

6.  4:a^  +  d¥  +  c^+12ab-4:ac-Qbc  7.  3^4^.2:^2  +  3+4  +  4 

X  X/ 

8.  4a;2  +  252/2  +  9c4  +  20a:i/-12c2^-30c2  2/ 

9.  -„  +  —  +6+     -^  +--^ 

10.  a;2  +  2/2  +  22  +  4_2x2/  +  2a;2— 4a;— 22/e  +  4y— 4^; 

11.  -2-2:c3  +  2;2/  +  2a;2/-22/3+^^ 
2/  "^ 

12.  m6  +  2m^  +  3m'*  +  4m3  +  3m2  +  2m+l 

13.  a3  +  ^,3  +  i^.3^2j.^3^2  +  3^j2  +  3j2.,.3^.,.3^^.(j^5 

14.  a;3-2/3— 23_3a;2y_3a;2  2  +  3a;2/2-32/22  +  3a:22_3^22  +  6a-y^ 

15.  a;3  +  8  +  2/3  +  6a^2.^3^2y  +  12a:  +  12y  +  3a;2/2  +  6?/2+i2xy 

16.  8a;3-272/3  +  125-36a;2  2/  +  60a;2  +  54a:2/^  +  135i/2+150a: 

-225?/ -180  a:  2/ 

17.  a:«  +  3a;5?/  +  6a-'4_iy2  +  7a;3^3_^g^2^4.,.3^.^5.^^6 

^«- ^7«^+ i^^+ 1^^+^^^4'^^^-^  1-^^+ 1^^^+ 1«^^+ 

25^  0      5     ^ 
^6c2+-a&c 

19.  2;3  +  82/3-2723  +  6a;2  2/-92;2  2  +  12xy2_362/2  2  +  27x22  +  542/22 

-36  a;  3/ -2 

20.  a;9  +  3x6  +  6a;3  +  7+  -o  +  -«  +  \ 

X^      x^      x^ 

21.  x«+6a^  +  9a;2-4--, +  ^--i 

a;2      a;^      a;^ 

22.  l  +  15a;  +  84a;2  +  215a;3  +  252a;4+i352.5^27a;« 

no      «      3    .      11    ^     17    ,     11    ,      2  8 

23.  2/«- 2-2/^+ -4  2/^- "8-2/^+-^/- 3  2/ +  27 

24.  a;6  +  3x5-12x4-29a;3  +  60a;'^  +  75a;-125 


5.  15 


Exercise  85. 

1.  ±18 

2. 

±36 

1              3.  ±48 

4.  8                5. 

6.  18 

7. 

±8 

8.  ±12 
Exercise  86. 

9.  12 

1.  ±ab^c^ 

2.  ±2a4j3c5 

3.  xy^z* 

4.  2mn^ 

5.  -Sx^y^ 

6.   ±mn^p* 

7o  -2aH^c^ 

8.   ±12a4a;3y5 

9.  -9(a+a;) 

ANSWJERS.  343 

10.  ±4(a-a-)«  11.  -(a  +  b)c^  12.  ±10x^x+y)* 

13.  -3(m  +  n)«  14.  ±2(a:»-/)2  15.   ±20244 

16.  ±21000  17.  1  18.  184 

19.1125  20.  ±1,    ±|,    ±f,  ±|,    ±1 

21.    ±r,,     ±-    r  ,     ±^-4t.     ±77  ^ 


22. 


2  1       a:  2.rgy3     g-^g  +  x) 

3'   ~  2  ^  y'   ~  3¥F'      c^rfs 


^(a-6)2'       (a;+y)3'    '^'(a-i)*'       2a» 


QA. 

2      2      3 

"^3'    3'    4' 

ax^        .1 

^(a 

-a-)3'    ^2 

Exercise  87. 

1. 

±(a  +  6) 

2.  ±(a:-y) 

3.  ±  (.r  +  4) 

4. 

±  ix-S) 

5.  ±{x  +  y  +  2) 

6.   ±{x-y-z) 

7. 

±(a+26  +  30 

8.  ±(2a:-3/  +  3) 

9.    ±{.c'  +  y+{) 

10. 

±{Sx-4y) 

11. 

-(^-^0 

12.  ±(2x'  +  ^y  + 

Exercise  88. 

1.  ±{x^+x+l)  2.  ±(a;«-2a:+l) 

3.  ±(a;»+2x+3)  4.  ±{x^-Sxy  +  2y^) 

6.  ±{3^-2x^  +  'Sx)  6.  ±{2x'^-5xy  +  3y*) 

7.  ±(^a:«+ia;+i)        8.  ±(^x'  +  2  +  ~'^       9.  ±  {x?^+x'-x  +  l) 

Exercise  89. 

1.  ±17  2.  ±20  3.  ±35  4.   ±43 

6.  ±52  6.  ±09  7.   ±71  8.   ±84 

9.  ±127  10.  ±245  11.  ±324  12.  ±408 

15.  ±.07  16.  ±.25  17.  ±.012  18.  ±29.7 

19.  ±.0004  20.  ±.324  21.  ±5.82  22.  ±3.38 

23.  ±10.42  24.  ±32.01  25.  ±9.999  26.  ±89.5 

27.  3.1022,  3.3106,  3.4641,  3.6055 

28.  1.4142,   1.8165,   .9354,   .5590 

29.  6.324,  6.403,  6.480,  6.557 


344  ELEMENTARY  ALGEBRA. 


1. 

x+\ 

4. 

2a:«+3 

7. 

ax—b 

10. 

-1 

Exercise  90. 

2.  a-6 

5.  x'^+x-\ 

8.  2aa;— 3&y 

11.  ax-  — 

ax 

13.  CH-5  +  C 

Exercise  91. 

2. 

27                     3.  35 

6. 

84                     7.  88 

10. 

222                 11.  305 

16. 

.12                  17.  .03 

20. 

.50                  21.  3.4. 

3.  rr  +  4 
6.  y^-y-\ 
9.  a;2-2a:  +  3 

12.  x^  +  l+K 


1.  14  2.  27  3.  35  4.  67 

5.  72  .           6.  84  7.  88  8.  98 

9.  122  10.  222  11.  305  12.  420 

15.  .2  16.  .12  17.  .03  18.  .25 
19.  .31  20.  .50  21.  3.4.  22.  2.4 

23.  4^  24.  1.442,  .854,  .646 

o 

Exercise  92. 

1.  ±{x^  +  y^  2.  ±(:x  +  y)  3.  ±4,  ±3,  5,  ±.2 

Exercise  93. 

1.  (a;2  +  «.r  +  a2)(a:2-aa:  +  a2)  2.  (a:  +  5)Cr  +  l) 

3.  (2,r  +  52/)(2a;+?/)  4.  {^x^  +  1  xy ^-4:y^){^x'^-l xy ■^4:y^) 

5.  {2p  +  q){p^Zq){2p-q){p-^q) 

6.  (8«'-^  +  4a6  +  962)(8a2-4(i&  +  962) 

7.  (.2;2  +  ic  +  3)(2:2  +  x-l)  8.  {2x■\-^y){x^-^J){2x'^  +  ^xy-y'^) 
9.  (2a;  +  l)(4a;2  +  28:r  +  61)  10.  (3a:-y)(9a:2-15a:2/  +  7y«) 

11.  (x  +  y){x—y){x^-^1lx^y^^-\^y^) 

12.  (2a2_3  62)(4a4  +  3  54)  13.  (a:  +  5)(a;2  +  25rK+175) 
14.  (2a3  +  i)(4a6_8a3  +  7)  15.  (^s  +  3  ^,3)  (^^e ^_ 3  ^,6) 

16.  (a2  +  2  6-^)  (a2_2  &2)  («8  +  «4  ^,4  +  7  ^,8) 

17.  (2a  +  46  +  5c)(2a  +  26  +  3c)      18.  (3a— &)(9a2— 42a6  +  61  62) 
1 9.  (a2 x-h'^y)  (a^ ;c2  +  4  ^2  i^  .r  2/  +  7  6^ y^) 

Exercise  94. 

1.  a;=±6  2.  a:=±2  3.  a-=±2  4.  a:=±i\/lO 


2 


6.  a;=±2V^  6.  a:=±l,    i^V^ 


ANSWERS.  345 

8.  x=±a/'_^^         9.x=±^a^        10.  a:=  ± >/6,   ± V^ 
11.  a;=±3  12.  a:=±\/2,    ±oV^  13.  a:=2,   -10 

14.  x=±V^iK^,   +l/_2—  15.  5  rd.  16.  15,   12 

10     "^  50 

17.  50  A.         18.  $90         19.  9,  21         20.  9  in.         21.  10  da. 


Exercise  95. 

1.  x=2,   -4 

2.  a:=6,    -4 

3.  x=-2,   -3 

4.  x=4,  5 

5.  x=G,   -3 

6.  a:=5,   -4 

7.  x=i,  7 

8.  a:=6,   -10 

9.  x=7,    -8 

10.  x-W,   -10 

11.  x=-l,   -li 

12.  a:=3,    -| 

13.  a=G,   -2^ 

14.  rr=3,   -4^ 

15..:=-7,    -1 

16.  j^=9,  -li 

17.  ^=-i|,  -4 

18.  ;r=7,   -2^ 

19.  ..-2^,-1 

20.  x=2^,   -Ij 

2           2 
21..=  -|,    -l| 

22.  a:=2i    -si 

23.  a:=2,    -3| 

24.  a:=3,    -2j 

25.  x=2±\^ 

26.  a;=3±2'v/5 

27.  2;=4,   i 

2S.x=6,   -1 

29.  a:=l±i\/2 

30.  a:=8,   -lo| 

3l.a-=|±iA/l3 

32.  a;=8,  3 
Exercise  96. 

1.  a;=a,   —3  a 

2.  x=2b,  b 

3.  a:=2m,    —3m 

a           a 

5.  x——ab,   —b 

6.:r=|(l±V-3) 

7.  a:=3a,    -a 

8.  x=|(1±a/5) 

9.  a:=&±  V«^ 

10.   2^=±\/i>? 

1                          

11.  a:=2a,    -« 

1 

12.  2:=a,  \ 

13.  T=l(-ft±2v^ 

-be)              1^^=2< 

[n±2^mn) 

15.  a:=~, 

a'        c 

16.0:=^ 

b 
a 

346 


ELEMENTARY  ALGEBRA, 


la     X 

4.  X-. 

7.  X 

10.  X 


=  ±2,    ± 


Exercise  97. 

^6  2.  a;=l,   2 

1 


3.  x=  ±  ^2,  ±  y^ij 

:±3,    ±2a/^         5.  a;=2,   ^  6.  x=l,    -2 

8.  a;=±7,    ±5  9.  a:=±l,    ±V^ 

11.  a;=-2±A/3,   -2±V-^ 


±1,    ±^ 


2a 


12.  a:=3,    -5 


13.  x=±2 


1     1 


14.  xz 

16.   CC: 

19.  a;: 
21.  X-- 


,0,  -1,  _1(1T  a/-1»)        15-  ^=3,    3,    4(-13±\/T53) 
=3,    -5,  2,    -4  17.  a;=7,   3,    ±2  18.  x=d,    -1 

:±1,    ±2  20.  x=j^{ym ±\/c +  171^— a) 

=  V2,    V^ 


22.  x=-2,   1,    -^  (IT  a/5) 
1      1 


23.  a,-=2,  -^,  -^(-7±a/33) 


2'    4 


=  ±4 

:3,     -6 

-i     -2 
-4' 


1.  X-. 

4.  a:: 

7.  x. 
10.  a;: 
13.  a;: 
15.  x: 
17.  a; 
19.  X 

21.  a:=±3,    ±3^/^ 
23.  X 


Exercise  98. 

2.  ^=±4 

5.  a;=— 2,   —5 
1 


8.  a;=2,   -1 
11.  x=5,   — 1 


3.  x=±^a 

6.  x=2,  4 

9.  a;=r-li     -1- 


12.  a;: 


2      3 
'3'    2 


14.  a;=2  g,   -5 


=2,  -1±V^ 
=  -2,  liV^ 
r-3,   i(3±V'^=27) 


16.  a;=-l,   _(l±V-3) 
18.  a;=3,  ^(-3±a/^^) 
20.  a:=±a,    ±a\/^ 
22.  x=±l,  ^{Tl±V^) 


±a,   ^a(Tl±A/-3) 


25.  a;=±2,    -6 


24.  :r=^-a,   |(-3±a/=27) 
26.  a;=±l,    +1 


ANSWERS.  347 

27.  2:=J(Tl±V^  28.  r=-l,    -1,    -1 

30.  a:=^(l±A/^) 

-2  32.  a:=2±A/3,  2±'v/^ 

33.  x=a,  b,  b—1 


Exercise  99. 

1.  x^-6x=-S  2.  x^-2xr=l5                3.  x^  +  5x=^24: 

4.  a:«  +  9a:=-20  5.  3;2-3aa:=-2a« 

6.  x*-px=Qp^  7.  a;«  +  7aa;=8a« 

8.  x*-2ax=b^-a^  9.  a;»-2a«a;=6*-a* 

10.  a:2-2a:=l  11.  a;«-6a;=-7 

12.  a;*— 4aa;=6«— 4a«  13.  x^—2ax=b-a^ 

%  A         9         1"  ^  tea        ^^  * 

16.  x^—2ax=4?n^—a* 

Exercise  100. 

1.  a;»-9=0  2.  a;2  +  2a;-35=0  3.  a;«-2a;-35=0 

4.  a:«  +  12a;  +  35=0  5.  a;»-12a;  +  35=0 

6.  12a:«-17x  +  6=0  7.  2a;»-5x-25=0 
8.  0a;«  +  13a:+6=0  9.  6a;«  +  13a:-15=0 

10.  x^  +  ax-2a^=0  11.  a:»-2a:«-9x  +  18=0 

12.  a^«-4a;«-9a;  +  36=0  13.  a:3-5a;«  +  8a;-4=0 
14.  a:3-4a:»  +  3a:=0                           15.  12a;3-4a:«-3a;  +  l=0 
16.  10a.-«-39a;«4-39a;-10=0           17.  16a;'-16a:*  +  3a:=0 

18.  60x»-133a;»  +  98a;-24=0 

Exercise  101. 

1.  x=^a(l±V2a*  +  l) 

4.  x=l,  I  5.  a:=-3,    -4 

7.  a:=±5  8.  a;=±9 
10.  a:=i(9±  a/145)       H-  a:=7,   ^ 

13.  a;=±'v/mn  14.  a;=-,   - 

c 


c^G,  6 

3.  a:=14, 

-10 

-4 

6.  x=±l 
9.  a:=21,  5 
12.  x=4,  0 

6 

15.  a;=a±  — 
a 

348  ELEMENTARY  ALGEBRA. 

16.  x=h,    -a         17.  a:=9.477,  -1.477         18.  a;=2.108,  -2.608 
19.  a;=5.236,   .764  20.  a:=1.148,    -0.348 

21.  a:=:±  1.095445  22.  a;  =±4.54923 

23.  a:=30.716,   -0.716  24.  a;= 7.464,  0.536 


2b.x=±a,    ±-  26.  x=±^-l,  ±Y^{l±^/-^) 


a 


27.  x=2,  i     i(-13±  Vl53)  28.  x=4:,   -3 


2'    4 
29.  x=l,    -1(1^^/^)  30.  a;=-l,  i(l±v^ 

31.    X=±l,      ±V^  32.    (x2-a2)(a;2_&2)(a,2_c2)^0 


Exercise  102. 

1.  12,  13  2.  3,  14  3.  7,  15 

4.  20  rows.  5.  30  yd.  6.  5,  25 

7.  8  rd.,   6  rd.  8.  40  mi.  an  hr.  9.  4  da.,   6  da. 

10.  5  hr.  11.  36  12.  12  ft. 

13.  $50  14.  $80,   $120  15.  $2000 

16.  3  in.  17.  20  ct.  18.  $24,   $30 

19.  2+^8  rai.  20.  24  mi.,  16  mi.  21.  6  ft,  4  ft. 

22.  $9  23.  $41.83  +  ,  $33.83+        24.  3  ft. 

25.  $2,   $3  26.  6,  8  27.  24,  18 


10. 

13. 
14. 


.    \  x=S,  4  I 
(  y=4,  3  f 


ANSWUBS.  849 


Exercise  104. 


2.    {x=5,   -4) 
ly=4,   -5f 
3.    ja:=3,   3,    -3,    -3^  4.    {x=2,  2,   -2,    -2 

iy=d,   -3,  3,   -3 
6.    {x=z4,   -4,  3,   -3  I  6.    \x=2,   -2,   1,   -1 

iy=3,   -3,  4,   -4)  (y=-h   1,   -2,  2 


1.    U=3,  2) 

I  y=2,  3  f 

a:=3,  3,    -3,    -3  } 

y=l,  -1,   1,   -1  f 


.  j  x=Q,  3  ) 
<  y=l,  2  f 


9. 


10.    ra:=3,   3,    -3,    -3       )        11.    r       A       1     _    1  1 

_1     _1    i    _l[              J'^-^3'     3'       ^3'  ^3 

^~2'       2'  2'       2)               1          1^    _1       J.  1 

12.    r        .1         .11          1^          '^^~  3'      ^2'  ^2;  ^2 


^=%'   -^2'    2'   -2 
(y=l,   -1,  3,   -3  > 


Exercise  105. 

x=±i)  2.   j.r=±3)  3.    {x=±2l 

y=±2)  (y=±S)  iy=±l) 

4.   \x=±Sl  6.    ja:=±2A/6{  6.   ja:=±5/ 

7.   ja;=±5)  8.   ja;=±5}  9.    {x=±4 

y=±3)  iy=±5)  (y=±l 


(y=±4^  iy=0  ) 

U=±5) 

iy=±5) 

(.=  ±1) 

iy=±l) 


10. 


3.   ja;=2,  5,    -4±\/G  }  4.    (a;=±2 

ly=5,   2,    _4T'v/G^  <2/=±1 

*•    fa;=3,  5^,   -6,   -3l|  ^'    (x=15,  10,  i(-23±  V^=^) 

[y=5,  2^,   -4,   -eij  |y=10,  15,  1(-23t  \/=^) 


350  ELEMENTARY  ALGEBRA. 


5.0-../— ^^f  <2/='J',   -3,   -1±2V 


y=6,   8,    ±|(3q:v^ 


Exercise  107. 

2/=  ±2,    T^a/sJ  [2/=  ±4,    ±^Vl3 


b2V'7_  J. 


5. 


-4  4^1       '■  h^^'^Vfl^ 


7.    r  .5     /— ir^  8.    ( a;=0,    ±2  ) 


Exercise  108. 

2.    ra:=7,   -18  ^  3.    r  .    o        ^3 


1.    (.T=5)  2.    ^a:=:7,    -18)  d.    r_3    _^±y 


y=±i(Va'  +  2&qF  V^'-^^) 


6.    {x=5,   1}  7.    jx=±5,    ±1)  8.    ja^=±3,    ±2 


2/=±l,    ±5i  <y=±2,    ±3 


ANSWERS.  351 

9.    (a;=3,    -2/  10.    ja;=±6)  11.   ja:=  +  8,    -8) 

(2/=2,   -d)  \y=±2,)  (2/=±4,    ±4) 

12.  (X=\,  2    ^  13.    (a;=±2,    ±3)  14.    ia:=±3) 
jy=i,    2^                     U=±3,    ±2f  U=±3y 

15.  U=±2/  16.    U=±5,    ±5V^)  17.    (a:=9      ) 
<2/=±2f                  (2/=±2,    ±2a/^J  (y=-^) 

18.  f  1  \  19.  U=5,  3,  -5,  -7) 
)^=^3n  l2/=3,  5,  -7,  -5) 
(y=±36   )                    21.    /a;=l,  3) 

20.    (a:=±5,    ±4)  ]  2/=2,  2^ 

<y=±4,    ±5f  (^=3,   l) 

Exercise    109. 

1.  5,  8  2.  3,   7  3.  80  rd.,  100  rd. 

4.  4  ft.,  6  ft,  5.  24  rd.,   10  rd. ;  26  rd.  6.  40,   $90. 

7.  8  da.  @  $8,  5  da  @  $5  8.  4,  9  9.  45 

10.  120,  $15  11.  16  rd.,  10  rd.  12.  10  ft,   12  ft. 

13.  $1.30,   $1.40  14.  4  mi.,  2  mi.  15.  22  ft. 

16.  15,  20  17.  9,  12,  15  18.  $7200,  $80,  $90 

19.  $66,  $20  20.  4  mi.,  3  mi.  21.  $100,  $100 


Exercise  110. 

1.  -2 

2.  - 

■8                   3.  -6|  yr. 

4.  -5,  45 

»•:? 

6.  - 

■45  yr.           7.  4,    -3 
Exercise  111. 

8.  ±8 

1.  4,    ±8,  243,  16 

2.  16,   625,  4 

3. 

1           11 
9'    '^'125'  256 

4.  256, 

±4, 

9 

5.  ±2,  9,  4,   16 

6. 

±64,000,  576 

7.  a-». 

arh , 

a\A 

8.  ax—W 

f,  ^- 

-^y-^ 

'•I- 

7* 

xzi 

11.  a  +  2a^bi  +  b,  a—2aibi  +  b,  a—b 

12.  x^ +  2 ziyHy^,  x^-2x^y^  +  y^,  ai-b-i 

13.  a;+3a;iyi+32?^yf+y,  x-*—Zx-^y-^  +  Sx-*y-*—y-*, 


ar-*  +  3x-^  +  Sx^-\-afi 


352 


14. 
16. 
18. 
20. 
21. 
22. 

23. 


ELEMENTARY  ALGEBRA. 

x—y  15.  x-"^ y^  +  x-^ y  +  xy-^—-x'^ y-^ 

x  +  y  17.  x^  +  x^yi  +  y^,   xr-^—y-^ 

xi-x\-\r\  19.  a^—2a^a^+a^,   (a;«-l)i 

{x^-\-x'^  +  \)\,  x-^+x-^y-^+xr-^y-^  +  y-^ 

{ai  +  bi)  (a^-bi),  (x^  +  yif 

(ai  +  bi)  (ai- &5 ),  (ai  +  bi)  {ai-ai  b\  +  b^), 

{a^-b^)(a^  +  aUT  +  bf) 
ai  +  4an3  +6 aibi  +  4ai¥  +  bf ,  a-^-ia-'^b-^  +  da-^b-^ 
-4  a-2  &-6  +  6-8,  a-^2^-5  at  + 10  at-10  a-f + 5  «-!-«-¥ 
1 


24.  1,  a«-6« 


(a-6)i 


1. 

3. 

5. 

6. 

7. 

8. 

9. 
11. 
12. 
13. 

15. 
17. 
19. 
20. 
21. 
22. 
23. 


Exercise  112. 

±2\/s,    ±S\/2,    ±4^/3"  2.  ±d\/5,    ±4:\/d,    ±5\/S 

2^,   -SVs;  4V2"  4.  -2^7,  3 V^  3^5" 

±2a^,    ±4:¥\/b,    ±2c^\^ 

±4:b\/a,    ±dx\^2xy,   —2aXfa 

±5ab^^^/2ab,    ±dxy^\/5y,  Zx\fx^ 

±4:a^b^\/ab,    —x^y^zXJxy'^z,    ±x^y z XJx z^ 

±a'\/a  +  b,   ±{a  +  b)\^        10.  ±(x  +  y)\^x(x  +  y),  {x-\-y)X/x 


-.{??-y'^Wx-y,  x{x  +  y)^x^y\x  +  y) 


±{x-\-y)^/x,  {x-vy^X/x^x^y)^ 

±2{x  +  y)^x—y  14.  ±ab (a—b) \/b (a^  +  ab  +  b^) 


±^v^,  i^Vs;  i 


^ 


16.  ±^V^,  |V9,  i. 


^100 


VIO,    ±1^30,   i 


180 


18.  ±|v^  iv^;  |V4 


2     / 1     / 1  Q  / — - 

±-=-y5xy,    ±  —  \^a{a  +  b), Z/ax 

oy  a  X 


'be,    ± 


36 


^15  a,   -=-^25  a 


±si6^«'+»*'  ±(^'V«'-*' 


ANSWERS, 

24.  ±l^/a:^-xy^    ±^a/^ 

X  X 

25.  Vx^{x-\-y\    ±- 7i'\/x—y 


35a 


26.  ±^^^^/^y,   i-y^ 


^     Exercise  113. 

I.  ±\/3i    ±2,    V3 
3.  ±2V^,    ±X/hx*y 
5.  ±— Vo^    ±— -x/— aa; 

7.  ±v^    a/s;    ±3 

9.  ±V^V^,   -3/V2^ 


2.  iiy^   ±^^6,    ±iV324 


11.  -3^Vl8^,    ±-V^ 


4.  ±yz^xz^    ±xy^/ay 

10.  V«-* 
12.  ±\/aVh 


Exercise  114. 


1.  V25,   V^,  |/f  a;* 


2.   V27a8,    yi256»^, 


3.  ^^^^^5,   V^v^,  i/|rl        4.  Vso,  VsT,  |/| 

7.  (a:*)!,  (y^A   8l  8.  (a:f  y)!,   (a^<^2)f  9.  (a;«y)«,   (2  a;)* 


1.  >^,  V9,   V^ 
3.  aW,  6A,  ctV 

*•  r   16'  y  729'  y  64 
7.  Vo*^.  'Vo^,  'V^« 


Exercise  115. 

2.  v^  v^,  v^ 


6.  V"^    V^    V«^ 


1.  12  Vo" 


2.|V2 


8.  Va^(a:+y)»,  (a;+y)A 

Exercise  116. 

3.  A/3a;  4.  6  V^" 


354 


ELEMENTARY  ALGEBRA. 


5. 
8. 

10. 

12. 
15. 
18. 


{a  +  l +  0)^/2 
\a       b       cj  ^ 


6.  2a\/x  7.  a  {'\/x  —  \/y  +  ^/^ 

^a^-b^ 


b{a—b) 
11.  {a^-b-c)^l 

13.  V^ 


2a^b 


19 


16.  2aya  or  — 2&\/a 
2a 


14.  f  ^ 

17.  2^4 


a2_62 


Y  a^  +  a  6 


20.  ^~—^/ax-x^ 


12. 
15. 
18. 
20. 


±3v^ 

1    / — 
c 


iv^ 

2    ^ 


18 
a—b 


Exercise  117. 

2.  Vl8  3.  ±^6"  4.  Tl2a\/2bc 

7.   ±2a\/6a  8.  G;^^^^ 

11.  Vsoo 

13.  TQ a ^aWc^  14.  2+^6'+ \/iO 

17.  Va-a; 


6.  IVs 


10.  ± n\/c{a—b) 

Cb U 


16.  Va^-64 

±abcxy'^^X/^^  19.  6\/3"-24  +  10 Vl5 

4x—9y  21.  x-y/x+y^/y  22.  x^  +  xy  +  y"^ 


Exercise  118. 


1. 
5. 

9. 
13. 
17. 
20. 


±2 

—  x/ac 
c  ^ 

±12 


2.  2v^ 


1 


3.^V5 

7.  2^6" 
11.  \/x  —  \^y 


4.|VT 
8.  aV^ 


10.  V4 
14.  12^/10  15.  ±63 

18.  V3"+3+a/5" 


'a—b^2b        21.  ^/x-\-^Jy 


12.  12^3 
16.  V^-l 
19.  3^2"+ 4-6  Vs" 
22.  x—^\/xy  +  y 


Exercise  119. 

1.  l/I  2.  9V^  3.  8  4.  108 

5.4^5"  6.  a^x/ab  l.Aa^b^  Q.  aH  ^/oFb 

9,ab{a  +  bf  10.  (a-b)^^^  11.  {x+yy 


12. 
15. 
18. 
21. 

23. 

1. 
6. 

9. 
13. 
16. 

19. 


6(5-2a/6) 


-^yV2x 


13 


ANSWERS. 
4 


355 


9 


14.  a  +  2\^alf  +  b 


16.  x—2\/xy  +  y  17. 


i» 


Vl2a& 


19.  p  20.  20+15^3" 

22.  2  +  3V3'-3V9' 

24.  a2(a-2V«^  +  6) 


Exercise  120. 

2.  V2a  3.  \/2a 

6.  Va*^  7.  Vs" 

11.  2!t/Sa 


10.  I^VSa 


14.  \^{a-\-b)x 
17.  \/a  +  6 

20.  ^j^^^W 


4.  ^^a^x 
8.  -v^ 

12.  Va^ 

15.  V60(a+a;) 
18.  3V2aa;« 

21.  -XJ^ 


1. 

4. 

7. 
10. 
12. 
16. 


2-V2 

1(11-6^2) 
a;+2Va;y+y 


Exercise  121. 

2.|V5 

6.  -r's/ab 

8.  -3(V2'+ v^) 


3.1^/15 
6.  -I-/V/2" 


a«-6 


11.  3  +  2V^ 


13.  2^-h 
16.  .7071 


14. 


a^—2a^/b^b 


a^-b 
17.  .1716 


Exercise  122. 

±3V^.    ±2aV— 1.    ±4V-1 
±5x^/^,    ±6aa;»V^»    ±7a«y'V^ 
±2v^xV^,    ±2V3a'xy^,    ±Sax^^/2x  X  ^/-i 
9^/^  5.  3aV^  ^-  -1,  -V^»  +1»  +\/-l 


356 


ELEMENTARY  ALGEBRA. 


7. 
10. 
12. 

14. 
16. 


-1,   +a/-1,   +1,  -V-l  9-  -6,  -10, 

-18,  -160,  -an^  11.  7,   1 

S^/~-i,  8  +  6V^,  -7  +  2^10 


V^ 


V3,  ±2,   ±- 


15.  ±4,  ±2x^/2^,  ±\\^ 


^/i     o    /' — o\    a^—h  +  2a^/—b     2's/ah—a—h 
_(l  +  2V-2),  -^^^ ,  j3^— 


1. 

4. 

7. 
10. 
13. 
15. 

17. 

19. 
22. 

25. 

27. 
30. 

33. 

35. 


1. 
4. 

7. 
10. 
13. 
16. 


Exercise  123. 

2.   ±(2-V5) 
5.  ±(3+V6) 
8.   ±(^5  2;  +  V^) 
11.  12.   ±{^x  +  y+^x—y) 


3.  ±(2+V3) 
6.  Not  a  square. 
9.  ±(V6'+V5) 


±(V3"+V3) 

±(y2~«  — -nA) 

±(V^-V6) 

±(2^2"- VT)  14.  ±(V'a;+22/+V^-22/) 

±(2v^_VlO)  16.  ±{^-^y) 

±(V^+1  +  aAJ  20.   ±(^6+ a/3)       21.   ±(2'v/3'-Vi1) 

±(Vl3+V3)  23.   ±(1-/^14)             24.   ±(l+A/a) 

±(|-Vl5  +  iV6")  26.   ±1(^15- VlO) 

±(2'\/3'+V^)  28.   ±(1  +  2V^)          29.  ±{^/^-^^) 


31.  ±(2\/5"-V'21)  32.  ±(1  +  V-1) 

34.   iCV^  +  .-r- V^— x) 
±{2^^—{x—y)\  36.  ±(V^  +  2+ V^-1) 


±i(Vl4+VlO) 


at  07 

(a4-6)t 


Exercise  124. 

2.  af  &f  c-^^ 
5.  2(:c-# 

tn 

8.  (a«— S")^ 


11. 


'a2j 


3.    Onb'n 


6.    a«— 1  Jn— 1 

9.  V^ 

12.  v^v 


14.  V(a^-«T 
17.   V^ 


15.  ^x{x  +  yf 
18.  Va«^ 


ANSWERS. 


357 


20.   V^ 

23.1/4 
5 

26.  (128)i 
29.  0)"* 
32.  (6)i 

35.  (aa;+a;*)— i 

38.  2V^3l4 
4 


21.   V^ 


41.  2 


v: 


-13- 


52.  (a    «t    )"• 


44.  2(2a-2  6)« 
47.  8(4a:y)i 

50.  a»(a-*a;«)i 


24.  \/{a+xy{a—x) 

27.  (32)1 

30.  (y-i)~^ 
33.  (l)-i 

36.  ix—^)p 
39.  2V^ 

45.  2(3a;+3y)» 
48.  4(ay)i 

51.  z^(xy^i 


55.  ± (a— 2^)^/0+^      56.  a*tya2 


54.  ±{a  +  b)\/a 
57.  -,  Va 


58.  ± r\/a^  +  ab 

a  +  o 


59.  ± -v/o^^ 


a—x 


CO.  x\/o^  61.  ±  -\\/a^-b^  62.  33  V^" 


63.  (y-2£)(a;2-3a)^ 
65.  ±i'V^«6T^V 
67.  y5"-V2"+V7' 


64.  (a  +  6)-2  V(a  +  &)* 
66.  V2'+V3'-V6' 
68.  a^+aibh  +  aibh  +  a^bi+ahbi  +  b^ 


-41 


70.  ^x\/lE 
o 


,X.iV4 


a'— aVT 


74.  4\/S+4\/2  75. 


72.  -g^VlOO 
4(1 +V5) 


a*— 6 

76.  ^(3V5'+3V3'+2yi0+2v^)  77.  -^(1  +  2^/^) 

78.  -4+ Vl5  79.  2V^    ^/T5,  3^3^ 

80.  VlO,    a/s;  2V2'  81.  2V^    V^    Vl5 

82.  \/2,  V^.    VS"  83.  (V«  + V^X^/a-V^ 

84.  (V^+2)(y^-2)  85.  (xi  +  yi)(a;i-yi) 


358 


ELEMENTARY  ALGEBRA. 


86.  {a^-\-yh){x^—y^ 
88.  (V-^  +  5)(V^-5) 

90.  (V^+  VsJcV^- V2j 

92.  a;f— a:i?/i  +  ?/i 


87.  (4+ Va;)(4-Va^) 
89.  (a;+ V5)(a^-V5) 
91.  :^;f +  a:ii/^  +  ^t 
93.  X5+x5y5+xi-yi  +  :Ay5+yi 

95.  4n;l-G:ci2/^  +  9yf 


98.  2:l  +  iC9  2/9+2/^ 


94.  xi—xiy^  +  xiyi—xiys+yi 

96,  8xi+12xiy\  +  18xiyi+27yi 

97.  8a;f-12ari2/i  +  18xi2/*-272/f 
99.   V^-VV         100.  a+V^  101.  a;(V^2_^ 
102.  v^-V^        103.  {^x  +  y){^x  +  ^y)         104 

105.  xi  +  {xy)^+y^  106.  2a^—x^  +  2aA/a^^ 

107.  a3_3aj2_(3a2  6_j3)^/ZY 

108.  a'^  +  {4ta  +  4:b)^/ab  +  6ab  +  b^ 

109.  a3_3^j2_(^,3_3a2^,)yC:Y 
111.  217-88 V'6  112.  9aV564 


3 


114.  ^V243^ 


116.  ±(^yx+i-^x-i) 

2  (x^-y) 


118. 


x^  +  y 


119. 


2a; 


121.  2a: -^/x^-1 

X 


110.  (100  + 18  V -2) 
113.  x^Q25y 
115.  ±(2  +  3^5) 
117.  ±(V^+V^ 

120.  -+2+^ 

122.  ±{x^  +  xiy^+yl) 


l.'a:=:±32 
5.  x=da^, 

8.  x=16 

11.  a;=4 


123.  xi  +  l  +  — 

xi 

Exercise  125. 

-218 


3.  x=±9 


2.  a;=214 
■7a^  6.  xz=±a^/±a^—l 


9.  a:: 


6 


15.  2;=15 


18. 


-r^y 


21.  .=1 


12.  x=l-r-  13.  a;=2 

4 

16.  a;=-5 
19.  a:=36 
22.  x=4  23.  a;=l 


17.  x= 


4.  .'r=±l 
7.  a:=29 

10.  x=^a 

o 

14.  2^=7 
1 


25 


20, 


-Kl^y 


'6-l\2 

24.  a;=a 


ANSWERS, 


359 


25.  x=S,   -  J^  26.  2=2,    -3  27.  x=2,    -1 


1 


28.  a:=::^(l±\/4a'+l)  29.  x=ia,   a  30.  x=a,   1—a 

32.  x=j{d\^a+l)^ 
35.  a:=4,  9 
37.  a:=21,  12 


31.  a;=-,-(l-a±\/l-<^«  +  «'^) 
33.  .?— rt  34.  a;=l,  0 

36.  j-=  I  ~(-6±a/4«c  +  6^)[^ 


38.  a:=5±2\/l3  39.  x=l,  ^Vl2  40.  a;=VX    V^ 

41.  2:=3,    -4^,   i(-3±V33)  4L2.  x=a,   a{l±'\/^) 


2'    4 
43.  a;=±3,   ±a/7,  ±a/-5,   ±a/^ 

45.  a:=25,   -9,  8 ±4^/29 

47.  x=4,  2/ =9 

49.  rr=17,  3/=8  50.  a:=l,  y=4 


44.  x=0,  6,  3(l±\/2) 

46.  x=2  ±  V^,   hi  ±  V-3) 

4  4 

48.  x=0,  ^ ;  2/=0,  ^j 

51.  x=Sy  16;  ?/=l,  9 

62.    (^=±5,    ±3a/^1,  etc,  53.  a:=l  +  V3;   -(3+^3) 


^^•=±0,    ±dv— ^>   ^i^^' 
<  y=±S,    tS-v/— 1»  etc. 


Exercise 

126. 

.8.  »  +  * 

a— ^» 

19.  x=5 

Exercise 

127. 

11       13      8      2: 
^'  5'  ^3'   15'  •'S'    9'   y 

2.  A,  a-x, 

a-6'  «^^ 

3.  1,  a;*+x«y«  +  y^,       *^ 

•^^      ^      ax  +  ay 

4.  1:3;  aft 

:1;  a:b 

6.  11  :  30;  3:5;  ;r«:  2;   o:c 

^"  2000 

7.  3  i               8.  The  first. 

^    be— ad 
5- :r- 

10.  a«- 

-&» 

11.  1  :2                       12.  1.414,   1.732 

13.  .707 

14.  Increased,  diminished. 


15.  Diminished,  increased. 


16.  $000,   $720  17.  20  yr.,  30  yr.  18.  6^  ft,  8^  ft 


14.  2a; :  5  :  :  1  :  2 


Exercise  128. 

15.  2x:dy  ::21o:206 


360  ELEMENTARY  ALGEBRA, 

Exercise  1S9. 


1.  x=% 

2.  rr=10 

3.  x=26               4.  x=^'- 

0 

...^4 

6.  x=da 

7.  x=\                 8.  a;=4 

9.  x=5 

xo.  .=«<*;;» 

11.    U=±13 

[ 

12. 

i;: 

=4)                  13.    \x=±Q) 

=2f                         U=±4f 

Exercise  130. 

1.  28  ft.,  21  ft.  2.  72  yr.,   60  yr.  3.  7,  6 

4.  150,000,   $30,000  5.  2000  sq.  rd. 

6.  $1022^,   11090  Y^  7.  50  mi.,  30  mi.  8.  8  cu.  ft. 

9.  ird,  or  3.1416xcZ  10.  Trr^,   16^,   16x3.1416 

12.  -^irr^,  288ir  13.  mx-^  A. 


11, 

4irr2, 

100  IT 

14. 

d^ 

gal. 

1. 

2a 

2.  ( 

7. 

2A/a 

11. 

wa«-i 

15. 

0 

3 


15.  ex— 5 


Exercise  131. 

3.  a  4.  00  5.  5  6.  oo 

8.  0  9.  1  10.  0 

12.  m  13.  0  14.  a 

16.  a  +  1  17.  1  18.  a* 


Exercise  132. 

1.  Z=40,   >S'=:220  2.  Z=5,  >Sr=192  3.  /=3|,  >S'=16i 

4.  ^=w,   S=hn^  +  n)  5.  Z=2r,  >S=r2  +  r 

7.  1=1,  d=-^  8.  /=43,   >S'=204 


6. 

d: 

=5 

9. 

1  = 

-4' 

n=\ 

21. 

1= 

=37,  n: 

=:10 

20.  a=l,   -i;  w=12,  15 
22.  w=ll,  a=5 


Exercise  133. 

1.  7,  10,  13  2.  $175  3.  $320  4.  7,  12,  17 

5.  5050  6.  5,  8,  11,  14  7.  1,  3,  5,  7,  etc. 

8.  14475  ft.  9.  8  da.  10.  $35.70  11.  1,  2,  3,  4,  5 


ANSWEKS. 

361 

12.  2,  5,  8,  11, 

etc.              13. 
16. 

3  da.,   10  da. 
$1500 

14.  2,   14 

Exercise  134. 

1.  4374 

2.  2916 

3.1 

^'      513 

6.  2»-' 

6.  4095 

-?^^ 

8.  2"-^ 

9.  96,   189 

10.  243,  5 

''•  m='^ 

1 

4^ 

=a 

12.  Z=2r»»;  >S'=2 (r"  +  r9 +  ;•«+...  .r)  19.2,  6,   18,  54,  etc. 


Exercise  135. 

1.  $429.98  2.  4  yr.         3.  $3200,  $1600,  $800,  $400,  $200 

8.  5^3,  5v^,  15V3  9.  1,  3,  9,  27 

10.  4,   16,  64  11.  6,   18,  54,  etc.        12.  2,  8,  32 


13. 

248,  842 

14.  3,  9,  27,  etc. 

15. 

(fo)' 

16. 

^_3y(x-i+rr« 

Exercise 

136. 

17. 

9,  27,  81; 

117 

1. 

3,   13^,   11-1, 

25 

«     aft 
2-  6-1' 

a«x 
'  a-] 

.'  a-6'  3/- 

^ 

3. 

5     124     192 
IV  999'   1111' 

11       2        11 

30'    165'  900 

4.  40  ft. 

6. 

300  rd. 

Exercise 

6.21^ 
137. 

min. 

1. 

94             2.  -4 

3.| 

4: 

7 
6«  + 

11^    21 
i2*  +  T2^ 

6. 

Ub  +  dc  +  d-a 

.\. 

31        22 
-20^+15" 

7.  8a 

8. 

{a—b  +  c)m^+{a  +  b—c)mn+  {b  +  c—a)n 

,9 

9.  -1 

10. 

S  +  2x-iyi+x- 

-lyi+2xhy- 

■i  +  .r*2/- 

1 

11.  X*- 

1 
256 

362 


ELEMENTARY  ALGEBRA. 


14. 
15. 

17. 
20. 

24. 

26. 
28. 
29. 
32. 
35. 
37. 
39. 
42. 
43. 

44. 
45. 
46. 
47. 

48. 

51. 
63. 
54. 


{x^  +  y^)  {a^—a^  y*  +  y^) 

(a^—x^  y^  +  y^)  {x^  +  xy  +  y^)  (x^—x  y  +  y^) 


(3a;  +  2)2(a:  +  3) 


^pq 


21.  1 


18. 


22. 


x—y 
1 


o?-aA-\ 


16.  a:  +  ; 

19,  ^+y  +  ^ 

' x—y+z 

23.  a;=l 


1  1 

^=3'  ^=4 


a;=4,  2/=^?  '2:=6  '25.  ic=^ 

x\y^z->t^{^Jxy—^xz—^/yz)  21.  7— Sy'S 

a3  +  6a2a;  +  6a2:2+a:3  +  (3a2  +  7aa:  +  3x2)  Vo^ 

31.  20 


x^  +  2x^y  +  y^ 

x=l~,    -3,    -^{iTVm 


30.  a:i^  +  4:c6  +  6+4  +  4i 


33.  a;=-&, 


a;"      a:;' 
-{a  +  c) 


34.  a;=±l 


36.  xt=±3,  y=±2 
38.  a;3— ica  +  l 


1 


a4  4-14a2i2  +  64 


x^—x^y\  +  yi 

a^—x^y^  +  xty^—xiv^  +  y^        40.  -^^         41.         ,  „     ,„,,     - 

amx^-\-  {a  n—h  m)  x^—{ap+b  n)  x  +  bp 

(a2  +  l)(:r+i)(x-l);   {a  +  2b){a-2b)(a+db) 

y^-2,  y^-dy,  y*-4:y'  +  2 

{2x+z){lQx^-8a^z  +  4:X^z^-2xz^  +  z^);  (x+\/^+y) 
8(^i:y_l)  {x-^/xy+y) 

(16«3a;2-24a3y2)|^   (xV^-xfif)^,   {aa^  +  bx^)—z 

1  49.  ^  50.  2;=  ±  a/^  -^(1 T  VS) 

a:=±^V2;  2/=  ±2  a/2" 
l+x  +  x^  +  x^+  etc.,   a:»,  a;* 


52.  ^^=±-^^265 


+ 


xy 


{x  +  yy 


55.  30 

57. 
69. 

60. 


56, 


x  +  y—2\/xy     1  3 


,  ±V9;  v^  -V-i 

rc=144  58.  2a;  +  7 

{m~n  +  r)y^  +  {m  +  n—r)  yz  +  {r+n—m)  2* 
x=-{^^a-l)^  61.   Vr         62.  2  63.  a;2-2a;=-2 


ANSWERS.  3Ga 

64.  x'*—{a  +  b  +  c  +  d)x^  +  iab  +  ac  +  ad  +  bc  +  bd  +  cd)x^— 

{ab  c  +  abd  +  acd  +  bcd)x  +  abcd=0 

65.  x=^  i-b±  ^12  a  c  +  b^)  66.  x=a,  {1  +  \/a)^ 

67.  a:=10,  6,  8±5a/5^  |-(25 ±  a/385),   i(25±V809), 

y=6,   10,   8t5V5^  ^(25^^^).   ^(25^^869) 

68.  y»  +  22/3  +  42/'^  +  3i/  +  3 

69.  a2.y4  +  (a-2a&)2/'  +  (i2-64-l)/  70.  ^=±"7^^^^ 

71.  a:=(a-2\/a)2  72.  :r=4,    1;  2/=l,   4 

73.  a:=27,   8;  y=8,   27_  74.  -r=2,  2/=3 

75.  a:=3,   5,    -5±V^^;  2/=^,   -"i,    -5ta/-26          _ 

76.  Odd,   odd,   even.  77.   +1,    -1,    ±1,    V±4 

78.  x=-\,  x-1 


Exercise  138. 

1.  $3600, 

$3200 

2.  15  yr.,  45  yr.           3.  48,  52 

4.  37 

-I 

6.  16  ft.,  4  ft.        7.  1  hr.  29  rain.  22^  sec. 

8.  10  da. 

9.  2663,   1662,  1000               10.  56  rain. 

.1.  Ig  mi. 

12 

.  27y^  rain.,  10—  min.,  or  43^^  rain.,  60  rain. 

.3.  24,  36 

14 

.  10  mi.                  15.  $2^,  50  ct. 

16.  pvp^—q^i  qyp^—ff 

18.  3  ct.  19.  60,  50,  40  20.  8  da.  21.  5  hr. 

22.  $1200  23.  20,000  24.  3,  4,  5  25.  36 

26.  $126,  $882  27.  2,  4,  6,  8,  10,   12,   14 

28.  180  lb.,   160  lb.  29.  8,  6  30.  $480 

31.  $7000,  $8000  32.  $225,  20^ 

910  454  413 

33.  10^^  da.,   14^  da.,   17j^  da.  34.  10  rain. 

yoU  16\3  oil 

36.  30  rai.  36.  3,  9,  27,  81  37.  4,  6,  8,   10,   12 

38.  i(l+ VS),  i(3+V^    39.  ^^\     seconds.    40.  9753 

41.  4,  6       42.  13  yr.        43.  64  cu.  ft,  512  cu.  ft. 
44.  2,  3,  4     45.  4840     46.  1  rd.     47.  10  ft,  12  ft. 
48.  $15,000        49.  12  rai.        50.  144  oz. 


SC)4:  ELEMENTARY  ALGEBRA. 

51.  2^  mi.,   2jj  mi.  52.  15^  53.  106-^  mi. 

54.  19  da.  55.  6  hr.,  3  hr.,  2  hr. 


APPENDIX. 

Exercise  1. 

1.  a;2+a;  +  l  2.  x'^-x—\  3.  2  a:— 1 

4.  5a;2-3  5.  2x^—^x  +  Q  6.  2a^  +  a  +  l 

7.  x  +  S  8.  2a;  +  3  9.  a:2  +  2a:  +  l 

10.  x-5  11.  2/  + 1  12.  a8-a4  54^58 

13.  a  +  b  +  c  14.  a;2  +  a:3/+/  15.  a^  +  1 

16.  a:-2y 

Exercise  2. 

1.  x  +  y                         2.  3;2  +  a;y  +  2/2  3.  2a;  +  l 

4.  3x-2                        5.  2a;  +  3  6.  2x+Q 

7.  a:2+a;  +  l  8.  dx—2 

Exercise  3« 

1.  2(3a:2  +  7rc  +  4)(4a;2  +  3a:-10)  2.  (a;2+a;  +  l)(a;+5)(2a:+l) 

3.  (a+l)(a2_a-l)(a2  +  a  +  l) 

4.  (a2-2a  +  3)(a2  +  2a-3)(7a  +  3) 

5.  (2a;+y)(a;2+a:3/+y2)(^2_^2/+2/2) 

6.  (a-2+a:i/  +  2^«)(a:  +  y)(a;+2y)  7.  (3a;2  +  2a:+l)(a:+l)(a;2  +  l) 

8.  (a;2  +  aa;  +  a2)(2a:-a)(3a;  +  a)        9.  (3a;2-4a:+2)(a;-3)  (2x4-3) 

10.  (2a4+5a2  +  3)(2a2-7)(2a2  +  7) 

11.  (5  22-1)  (422^1)  (522  +  2+1) 

12.  (Sx  +  2)(x^-x^  +  x-l){a^  +  x^+x  +  l) 

13.  (3a:  +  4)(2a:3  +  3a;2-4a:  +  2)(3a;3  +  2a;2-3a:  +  l) 

14.  (2a2_6a-7)(6a'-lla2_37a_20) 

15.  3(w  +  3)(2m2-m  +  l)(3m2-7w  +  4) 

16.  (n-2 a) {n^  +  an^  +  a^n  +  a^)  (3  w^-a n  +  a^ 


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